Mickaël Dievart
email: mickael.dievart@enit.fr

Philippe Charbonnaud
email: philippe.charbonnaud@enit.fr

Xavier Desforges
email: xavier.desforges@enit.fr

Applicative architecture for embedded distributed technical diagnosis

Keywords: Distributed diagnosis, health status, multi-agent system, complex systems, embedded systems, solving method, system architecture, system diagnosis

This article presents an applicative architecture based on a solving method for embedded technical diagnosis of complex systems. This architecture is defined in order to provide services enabling the evaluation of the health status of complex systems. Diagnostic services provide information to the maintenance decision support system that leads to reduce the periods of unavailability and determine if their future mission can be carried out. The architecture presented in this paper implements a distributed diagnostic function using multi-agent techniques. A consistency model-based diagnosis is proposed that leads to the identification of the faulty LRUs and the failed functions of complex systems.

INTRODUCTION

For transportation systems, new regulations in terms of environment, goods and people protection, and needs of new services have consequences on the complexity of embedded systems. To face this increasing complexity, multiple functionalities of the resources are embedded and deployed into networks of functions achieved by Line Replaceable Units (LRU). Faulty LRUs are replaced when the vehicle is at its base and repaired in the maintenance workshops, while the repaired system carries on with its mission. The increasing number of functionalities of the embedded systems contributes to raise the possession and acquisition costs leading the resources customers to optimize their availability rate.

Using the Condition-Based Maintenance (CBM) recommendations usually improve the equipment availability [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF][START_REF] Scarf | A Framework for Condition Monitoring and Condition Based Maintenance[END_REF]). Indeed, the CBM depends on the effectiveness of the system state provided by monitoring and diagnostic functions. They are carried out in particular from on line data generally processed by an embedded centralized diagnosis function. However, in the case of system of systems also called complex system, the identification of the faulty components is difficult using centralized architectures. After the mission, the maintenance operators must collect information by interactions with the embedded diagnostic system, in order to isolate possible faulty LRUs, and to apply troubleshooting procedures. The drawbacks of such architectures are related to the numerous pieces of information to process, which might be wrong. The automated diagnostic processes combine these errors and lead to useless removals of LRUs. Those removals are costly and increase the risk of damaging the system.

Alternatively, a decentralized/distributed diagnosis can be proposed to reduce the number of useless removals of LRUs. For applications to system of systems, monitoring and diagnostic functions can be implemented closer to the LRUs thanks to agents that carry out them. In the case of a distributed approach, a collaborative mechanism between diagnostic agents have to enable the convergence of the local diagnoses towards a set of accused LRUs which should ideally correspond to the true faulty ones. This article presents an applicative architecture for implementing a distributed diagnostic function. In section 2, the problem statement is established. In section 3, the difficulties of implementing this diagnostic function in such systems due to the various kinds of the subsystems and to the necessary knowledge and models for its achievement is discussed. In section 4, an embedded diagnosis function is proposed. In section 5, an applicative architecture and its cooperation protocol is presented. Its objective is to carry out the identification of a set of faulty LRUs from LRUs declared faulty by the local diagnoses.

PROBLEM STATEMENT

The technical diagnosis of transportation systems provides to the maintenance operators a list of LRUs that should be replaced. When the diagnosis is done online, the maintenance operators prepare the intervention sooner, reducing the duration and the costs of maintenance actions. The main task of a diagnostic function is to deliver an advice on a set of faulty components and to determine the severity of the fault. A difficulty in diagnosing such a complex system is due to their numerous kinds of functions integrating different technologies (electronics, data processing, mechanics, hydraulics…). Thus, the implemented diagnostic techniques must be adapted to the knowledge available about the system. During its use, various faults may impact the resource. Those faults degrade its operating modes. Three types of faults are considered:

Cataleptic, the failure of the system is immediate. The fault implies a cataleptic failure. The system is in total breakdown. The faults can be simple, multiple or hidden, Permanent is a state in which one or several system functions are in degraded mode. The faults did not involve the failure of the system but degrade its performances and make it unable to fulfill all its objectives. The degradation of performance can involve the system locally (a function for example) or the whole system, Fugitive, the system switches between a nominal operating mode and a degraded one. This mode implies the same problems as those quoted previously. This type of failure can be not signaled or not explained.

In figure 1, a typology is displayed showing the multiple faults and their consequences on the health status of the equipment. The diagnosis, according to the various types of faults (typology) occurring in the system, classify the failure and determine the operating mode of the system (normal mode, degradated mode, exception mode…). A FMEA (Failure Modes, Effects and Analysis) can address this problem by the construction of causal trees.

Different implementation of the monitoring and diagnosis functions associated to the LRUs, are described in figure 2. The LRU1 hosts its monitoring and diagnostic functions, the diagnostic function can also be hosted in another platform (Diagnosis2) or, the monitoring and diagnosis functions can totally be distributed in different platforms as for Diagnosis3 and Monitoring3.

Fig. 2. Structure of a networked embedded supervised system.

In the case of aircrafts, the Centralized Maintenance System (CMS), provide a list of likely faulty LRUs for the maintenance operator. This list is established according to information from the built-in test equipments that collect information from the LRUs and generate tests if needed. The CMS correlates data to provide a "pre-diagnosis" of the LRUs. The flight warning system provides to the cockpit crew information on aircraft failed functions [START_REF] Ramohalli | The honeywell on-board diagnostic and maintenance system for the boeing 777[END_REF][START_REF] Byington | Embedded diagnostic/prognostic reasoning and information continuity for improved avionics maintenance[END_REF].

Complex systems can be considered as sets of systems that depend more or less on each other. A system implements one or several functions. For safety purposes, functions can be redundant as well as the LRUs that implement them. That is why several models are necessary to classify the different operating modes of the LRUs and their health status.

SYSTEM OF SYSTEMS MODELING

Generally, the system is analyzed from different models in order to obtain a satisfying representation for diagnosis purpose. This analysis enables to collect the available knowledge on the complex systems. Many of them are helpful to design their diagnostic functions. In the literature, these models can be functional [START_REF] Abu-Hanna | Device understanding and modeling for diagnosis[END_REF], structural and behavioral [START_REF] Chittaro | Hierarchical model-based diagnosis based on structural abstraction[END_REF][START_REF] Keuneke | Device representation-the significance of functional knowledge[END_REF], teleological [START_REF] Chittaro | Functional and teleological knowledge in the multimodeling approach for reasoning about physical systems: a case study in diagnosis, Systems, Man and Cybernetics[END_REF]. They enable to model the behavior of components, of functions and of their interactions according to normal or degraded modes. In other studies, models are added to evaluate the diagnosis confidence [START_REF] Bonarini | Uncertainty and approximation in multimodel diagnosis[END_REF].

A Complex System (CS) can be defined by a finite set of m system ∑ i . CS = {∑ 1 , ∑ 2 ,.., ∑ m }. A system ∑ i can be defined as a set of n function F i,j . ∑ i = {F i,1 , F i,2 ,..,F i,n }. A function F i,j can be defined as a set of k LRUs implementing this function.

F i,j = {LRU i,j,1 , …,LRU i,j,k }.
If a LRU contribute to the implementation of more than one function, a decision has to be taken when defining the system. A LRU should be part of one and only one function. After defining the hierarchical decomposition of the system, a system modeling has to be formalized to ensure system diagnostic.

In this paper, the set of necessary System Knowledge to diagnose the complex system is collected in the set SK made of four types of knowledge: functional, structural, behavioral and topological. The Functional Description (FD) is the set of functions ensured by every system. FD represents links between LRUs, functions and system. The Structural Description (SD) is dedicated to the identification of the set of LRUs and of physical connections between them. SD introduce predicate CONNECT(X,Y) that means that X is connected to Y. S : CONNECT(LRU i,j,q , LRU p,r,s ) with q and s respectively one of the LRUs implementing F i,j and F p,r . The behavioral models are used in order to identify the relevant indicators that are used to generate symptoms for the various faults that may affect the LRUs. They help to classify the faults of the LRUs from their symptoms. For diagnostic purpose, the knowledge BM provides the relationships between the symptoms, the LRUs and their faults. The Topological Dependencies (TD) determines the proximities of components that may be the origin of indirect failures or faults of a LRU due to the proximity of another failed one. TD introduce predicate TOPO(X,Y) that means that X is close to Y and that some faults of X may affect the functioning of Y. TD: TOPO(LRU i,j,q , LRU p,r,s ) with q and s respectively one of the LRUs implementing F i,j and F p,r . Finally,

BM TD FD SD SK    

EMBEDDED TECHNICAL DIAGNOSIS

In this paper, the diagnostic function consists of several activities: to condition, to detect and to identify failures and their causes. Relevant and significant indicators are generated by the function "to condition" and are based on measurements of the system. These indicators can be statistics (mean, standard deviation, …), signals values (magnitude, power, frequency, …), parameters (structural or physical parameters), state observers, residuals, errors, etc. The function "to detect symptoms" uses these indicators to generate values, called "detected symptoms", which are provided when a fault occurs. The decision can be made thanks to a decision-making support technique. The outputs can be digital and define which fault occurred. "To identify failures" is done by two sub-functions: "To identify failed function(s)" and "To identify faulty component(s)". The user (pilot, driver, …) cares about the failed functions of the system whereas the maintenance operator cares about the faulty components that must be replaced. In figure 3, the activity diagram of the function "To identify failure" is detailed.

APPLICATIVE ARCHITECTURE

Distributed

approaches of Information and Communication Technologies (ICT) often provide good enough solutions to face complexity. The diagnostic function was implemented in a distributed structure according to the multi-agent system concept. The agents of the structure cooperate and exchange data whatever the language used to model the information they contain is. This implementation requires data and models that have been collected and organized. In the case of the complex systems some works show the feasibility to implement an embedded diagnostic function by distributed architectures with or without cooperation between its elements [START_REF] Biteus | Distributed Diagnosis and Simulation Based Residual Generators[END_REF][START_REF] Heck | A Multi-Agent Based Monitoring and Diagnosis System for Industrial Components[END_REF][START_REF] Wörn | Multi-Agent Architecture for Monitoring and Diagnosing Complex Systems, The Fourth International Workshop on Computer Science and Information Technologies[END_REF].

The conceptual framework of the proposed applicative architecture is depicted in figure 4, the architecture presented herein is based on a distributed implementation. The local diagnosis agents cooperate to provide the diagnosis of the system. This implies the use of multi-agent system techniques. As shown in figure 4, a facilitator agent can be introduced, to ensure the convergence toward a solution within a given time. The applicative architecture does not imply the implementation of a distributed diagnostic function. It depends on the "solving method". The middleware solution eases the implementation of the services provided by the agents that are software entities exchanging data by abstracting the way they are hosted on different hardware entities.

The architecture consists of several LRUs that are gathered into several functions which is the LRU layer. Each LRU is observed by a monitoring function designed by the supplier. The monitoring functions are represented by the monitoring layer in figure 4. Then, the monitoring agents send their symptoms to a Diagnostic Agent (DA), which is in charge of elaborating the set of faulty LRUs. One or more databases (KB) contain the structural, the topological, the functional and the behavioral knowledge in order to provide a support for the different agents. A Human/Machine Interface (HMI) ensures the information displayed to users. The HMI displays, according to the type of operators, the failed functions of the system (for the production operators) or the LRUs that need to be replaced or fixed (for the maintenance operators). If the collaboration is correct the global diagnosis of the system in terms of faulty LRUs is a union of each local diagnosis.

The solving method of the distributed architecture is represented by the activity diagram shown in figure 5. These activities are carried out by cooperation between the DAs and the HMI. This cooperation is represented by sequence diagrams shown in figures 6 and 7. The reception of a symptom by a DA launches the process. The corresponding DA begins by defining the symptom received. If the symptom is generated by the monitoring layer, it is declared as a "failure symptom" and if it is a symptom generated from the function "To propagate symptom", it is called a "propagation symptom". If the symptom is a failure symptom, this one is diagnosed as described by the sequence diagram in figure 6 where the DA sends a request to the database to know if the symptom is known or unknown. A known symptom is a symptom for which the cause is already identified by studies made at the system design stage (FMEA). If the symptom is known, the database returns the result and the DA declares the cause of failure of the LRU as known, otherwise, the cause of failure of the LRU is declared as unknown. Then the fault is propagated as described by the sequence diagram shown in figure 7. This activity begins by a request of the DA to the database to know if it exists structural dependencies with other LRUs. If it is the case, "propagation symptoms" are sent to advice the DA in charge of diagnosis of the involved LRUs that they may not operate correctly. Therefore, those LRUs are declared as out of order. Furthermore, the supervision of the diagnostic agents is ensured during a cooperation task and more precisely during "the fault propagation" function. If an agent did not confirm that it receives the message during the propagation task, the diagnostic process accuses and declares the agent as failed. So, the diagnosis of the DA also contains information about the DAs that did not answer the request. By this way, the diagnostic of the DAs is carried out.

Every DA of the architecture diagnoses LRUs implementing a function. Every time a DA receives a symptom the last diagnosis is copied and updated according to the new received symptom. This ensures the elaboration of an historic of the evolution of the diagnostic process. This enables a non monotone diagnostic process.

Fig. 4. Conceptual framework of applicative architecture for embedded distributed diagnosis

If the symptom received by the DA is a "propagation symptom", this one is used for a "diagnosis refinement" task. To explain the activity, let us considered the example composed of 3 LRUs (

LRU , 2 , 1 , 1 LRU ), CONNECT( 2 , 1 , 1 LRU , 3 , 1 , 1
LRU ). Each LRU has its own monitoring period for each of its symptom. For example, if we consider one symptom per LRU, we suppose that the symptom emitted from the monitoring of LRU is updated to a out of order status. Finally, the DA Structural dependencies between LRUs are described in the set SD which is the only considered knowledge. The knowledge is recorded in XML files, which is an input of the solving method for symptom generation and diagnostic support.

Fig. 8. Case study of a monitored system of systems.

Four different status describe the state of a LRU:"OK" where the LRU is not faulty at all, "UF" when the LRU is faulty but the cause is unknown (Unknown Failure), "KF" when the cause of the failure is known (Known Failure) or "OO" when the LRU does not work in nominal mode or is failed because of the failure of a LRU structurally dependent of it (Out of Order).

The symptoms received by DAs are recorded in a field linked to the corresponding LRU. The state of the LRU changes and the diagnostic timestamp of the LRU is updated with the current one. At the end of the symptom generation session, a list of faulty LRUs with its cause and a list of failed functions are available. The cause of faulty LRU is described in terms of sentences from FMEA studies or of faulty LRU in the case of "OO" state. The cause of failed function is described in terms of faulty LRUs.

For example, if LRU 2,1,1 and LRU 3,1,1 failed and the cause of the failure of LRU 2,1,1 is known to be a power failure and the cause of LRU 3,1,1 unknown. After symptom generation, the diagnosis result is given by: In each set ∆ 1 and ∆ 2 is listed the LRUs or functions that do not work in a nominal mode with their current state, the cause of their failure and the timestamp their state changed.

∆ 1 = LRU 2,
All pieces of diagnostic data and their timestamp are recorded to ensure performance evaluation at the end of the diagnostic process. Performance indicators are therefore defined. These indicators use the timestamps of the different data that are exchanged between the agents on the middleware to be evaluated like: speed of convergence, data flow, and computational load.

CONCLUSION

The proposed distributed diagnostic architecture provides a solution to diagnose faulty LRUs and failed functions in the case of system of systems. This architecture is based on DAs that receive symptoms from monitoring layer. The proposed distributed implementation of the technical diagnostic function is based on a distributed resolution method and system knowledge. Future work will deal with the comparison between decentralized and centralized diagnostic functions.

Fig. 1 .

 1 Fig. 1. Considered fault typology and consequences on status.

Fig. 3 .

 3 Fig. 3. Activity diagram A3 of the function "To identify failure".

  are declared as out of order.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Activity diagram of the considered solving method.

  LRU should be defined as out of order. But the symptoms are not received in the order they occur because of the different monitoring periods. Considering symptoms

																		1 , 1 , 1 LRU can be
	send to the DA every 5 minutes, the one of	2 LRU every , 1 , 1
	minute and the one of			3 LRU every second. In such a case, , 1 , 1
	we consider that only		1 , 1 , 1 LRU failed,	2 LRU and , 1 , 1
				, 1 , 1	3
	t i S , t k j , ,		of the	j LRU , i ,	q	that occurs at time t and
	1 t		t	2		t	3	it can be infer that	1 1 , 1 , 1 , 1 t S		2 , 1 , 1 t S	2	1 ,		3 1 , , 1 t S	1 , 3	. It
	is important to make a difference between the time the
	symptom occurs and the time the symptom is send. At the
	first step, because of the monitoring periods, the DA
	receives	3 1 , , 1 t S	3	1 ,	. So,	3 LRU is declared has failed. Then, , 1 , 1
	the DA receives	2 , 1 , 1 t S	2	1 ,	. As	2 t t 	3	and because of the
	structural dependencies,	2 LRU is declared failed and , 1 , 1
				, 1 , 1	3

  1,1 (status: KF, cause: power failure, timestamp: 2009/07/05 15h26min56s) & LRU 3,1,1 (status: UF, cause: unknown, timestamp: 2009/07/05 15h27min05s) & LRU 1,1,2 (status: HS, cause LRU 2,1,1 failure, timestamp: 2009/07/05 15h26min57s) & LRU 3,1,2 (status: HS, cause: LRU 3,1,1 failure, timestamp: 2009/07/05 15h27min06s)

	∆ 2 = Function 11 (status: HS, cause: function 12 failure,
	timestamp: 2009/07/05 15h26min58s) & Function 21 (status:
	KF, cause: LRU 2,1,1 failure, timestamp: 2009/07/05
	15h26min57s) & Function 31 (status: KF, cause: LRU 3,1,1
	failure, timestamp: 2009/07/05 15h27min06s)

Acknowledgement: The project was supported by Airbus France


Jean-Paul Daniel
email: daniel@ann.jussieu.fr

QUADRATIC EXPANSIONS AND PARTIAL REGULARITY FOR FULLY NONLINEAR UNIFORMLY PARABOLIC EQUATIONS

Keywords: Mathematics Subject Classification. 35B65, 35K55, 35D40, 49N60 partial regularity, fully nonlinear parabolic equations, viscosity solutions

published or not. The documents may come

Quadratic expansions and partial regularity for fully nonlinear uniformly parabolic equations

1. Introduction 1.1. Motivation and statement. In this paper, we prove a partial regularity result for viscosity solutions of the uniformly parabolic equation

∂ t u + F (D 2 u) = 0 in Ω ⊆ R d × (-1, ∞). (1.1)
We write u as a function of (x, t) ∈ R d × [-1, ∞) and D 2 u denotes the Hessian of u with respect to the x variable. The operator F is assumed to be uniformly elliptic and to have uniformly continuous first derivatives. Recently, Armstrong, Silvestre and Smart [START_REF] Armstrong | Partial regularity of solutions of fully nonlinear, uniformly elliptic equations[END_REF] obtained a partial regularity result for viscosity solutions of the uniformly elliptic equation

F (D 2 u) = 0 in U ⊆ R d , (1.2)
with the same hypotheses on F . Specifically, they proved that, for every 0 < α < 1, a viscosity solution of (1.2) is C 2,α on the complement of a closed set of Hausdorff dimension strictly less than d.

We extend this result to the parabolic setting by showing that the singular set of a solution of (1.1) has Hausdorff dimension at most d+1-ε, where the constant ε > 0 depends only on the ellipticity of F and d. The hypotheses (F1) and (F2) are given in the next section. In this paper, Hölder spaces such as C 2,α are to be understood in the parabolic sense (e.g., a parabolic C 2,α function may only be C 1,α/2 in the time variable: see Section 2.1 for the precise definitions).

Theorem 1.1. Assume that F satisfies (F1) and (F2). Let u ∈ C(Ω) be a viscosity solution of (1.1) in a domain Ω ⊆ R d+1 . Then there exist a constant ε > 0, depending only on d, λ, Λ and a closed subset Σ ⊆ Ω of Hausdorff dimension at most d + 1ε, such that, for every 0 < α < 1, the solution u belongs to C 2,α (Ω \ Σ).

A new difficulty arising in the proof of Theorem 1.1 in the parabolic setting is to obtain an analogue of the W 3,ε estimate, an important and useful tool from the regularity theory of nondivergence form elliptic equations [START_REF] Lin | Second derivative L p -estimates for elliptic equations of nondivergent type[END_REF][START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF] (see also [START_REF] Armstrong | Partial regularity of solutions of fully nonlinear, uniformly elliptic equations[END_REF]). We prove it for viscosity solutions of the uniformly parabolic equation

∂ t u + F (D 2 u) = g in Ω ⊆ R d × (-1, ∞), (1.3)
where g ∈ C 0,1 (Ω). To give the precise statement of this result, we require some notation. We denote by M d the set of real d × d matrices. The open ball of R d centered at x of radius ρ is denoted by B ρ (x). If x = 0, we simply write B ρ . The following elementary cylindrical domains play a central role in the theory: for all ρ > 0 and x ∈ R d , we define Q ρ (x, t) := B ρ (x)×(t-ρ 2 , t) and denote Q ρ := Q ρ (0, 0). Now we define, for u : Ω → R, the quantity

Ψ(u, Ω)(x, t) := inf A ≥ 0 : ∃(b, p, M ) ∈ R × R d × M d s.t. ∀(y, s) ∈ Ω, s ≤ t,
u(y, s)u(x, t)p • (yx)b(st) -1 2 (yx) • M (yx) ≤ 1 6 A |x -y| 3 + |s -t| 3/2 , which represents the cubic error in the best quadratic approximation of u at (x, t). We emphasize this error is measured globally in Ω ∩ {(y, s) : s ≤ t} and, of course, "cubic" in the time variable means cubic in t 1/2 .

The statement of the parabolic W 3,ε estimate is given by the following theorem.

Theorem 1.2 (Parabolic W 3,ε estimate). Assume F satisfies (F1), g ∈ C 0,1 (Q 1 ) and u ∈ C(Q 1 ) solves (1.3) in Q 1 . Then there exist universal constants C, ε > 0 such that, for all κ > 0,

(x, t) ∈ Q 1/2 0, -1 4 : Ψ(u, Q 3/4 )(x, t) > κ ≤ C κ sup Q1 |u| + |F (0)| + g C 0,1 (Q1) -ε .
The elliptic analogue of Theorem 1.2 has been used for example to obtain quantitative estimates for the convergence of monotone finite difference schemes [START_REF] Caffarelli | A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs[END_REF] as well as rates of convergence in homogenization [START_REF] Armstrong | Quantitative stochastic homogenization of elliptic equations in nondivergence form[END_REF][START_REF] Caffarelli | Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media[END_REF]. We expect Theorem 1.2 to have similar applications in the parabolic setting.

The overall idea of the proof of Theorem 1.2 is similar to the elliptic case: we differentiate the equation to obtain the result from the parabolic W 2,ε estimate. In the parabolic case, there is an extra difficulty in controlling the derivative with respect to time. Unlike the elliptic case, here we need to use the PDE once more in order to show that by controlling all of the spatial derivatives we gain control over ∂ t u.

The argument for the partial regularity result is similar to the idea outlined in [START_REF] Armstrong | Partial regularity of solutions of fully nonlinear, uniformly elliptic equations[END_REF]. We apply a result of Wang [START_REF] Wang | Small perturbation solutions for parabolic equations[END_REF], which asserts that any viscosity solution of (1.1) which is sufficiently close to a quadratic polynomial must be C 2,α . This result is a generalization of a result of Savin [START_REF] Savin | Small perturbation solutions for elliptic equations[END_REF] in the elliptic setting. Theorem 1.2 gives us such quadratic expansions except on a set of lower parabolic Hausdorff dimension.

Structure of the article:

We start by gathering our notation and some preliminary results in Sections 2.1 and 2.2. The proof of Theorem 1.1 is presented in Section 2.3. Section 3 is devoted to the derivation of the parabolic W 2,ε estimate. Section 4 gives the proof of Theorem 1.2. A convenient way to write the Pucci operators is

Preliminaries and proof of the partial regularity result

P + λ,Λ (M ) = -λ µj >0 µ j -Λ µj <0 µ j and P - λ,Λ (M ) = -Λ µj>0 µ j -λ µj <0 µ j ,
where µ 1 , . . . , µ d are the eigenvalues of M . These operators satisfy the inequalities P - λ,Λ (M ) + P - λ,Λ (N ) ≤ P - λ,Λ (M + N ) ≤ P - λ,Λ (M ) + P + λ,Λ (N ) ≤ P + λ,Λ (M + N ) ≤ P + λ,Λ (M ) + P + λ,Λ (N ). For the modern theory of nonlinear parabolic equations we refer to [START_REF] Nicolai | Nonlinear elliptic and parabolic equations of the second order[END_REF][START_REF] Gary | Second order parabolic differential equations[END_REF]. A nice introduction to viscosity solutions of nonlinear parabolic equations can be found in [START_REF] Imbert | Introduction to fully nonlinear parabolic equations[END_REF]. Let Q ⊆ U ×(0, T ) and α ∈ (0, 1]. The set of upper and lower semicontinuous functions on Q are denoted by USC(Q) and LSC(Q), respectively. We will use the following notation.

• u ∈ C 0,α (Q) means that there exists C > 0 such that for all (x, t), (y, s) ∈ Q, we have

|u(x, t) -u(y, s)| ≤ C |x -y| α + |t -s| α 2
.

In other words, u is α 2 -Hölder continuous in t and Du is α-Hölder continuous in x.

• u ∈ C 1,α (Q) means that u is α+1 2 -Hölder continuous in t and Du is α-Hölder continuous in x. • u ∈ C 2,α (Q) means that ∂ t u is α 2 -Hölder continuous in t and D 2 u is α- Hölder continuous in x.
Throughout this paper, the nonlinear elliptic operator F : S d → R satisfies each of the two following conditions: (F1) F is uniformly elliptic and Lipschitz; precisely, we assume that there exist constants 0 < λ ≤ Λ such that, for every M, N ∈ S d ,

P - λ,Λ (M -N ) ≤ F (M ) -F (N ) ≤ P + λ,Λ (M -N ).
(F2) F is C 1 and its derivative DF is uniformly continuous, that is, there exists an increasing continuous function ω

: [0, ∞) → [0, ∞) such that ω(0) = 0 and for every M, N ∈ S d , |DF (M ) -DF (N )| ≤ ω(|M -N |).
We call a constant universal if it depends only on the dimension d and the ellipticity constants λ and Λ. If U ⊆ R d and s < t, then the parabolic boundary of U × (s, t) is denoted by

∂ p (U × (s, t)) := U × {s} ∪ ∂U × (s, t).
It is convenient to work with the parabolic Hausdorff dimension of a set E ⊆ R d+1 , which is defined by H par (E) := inf 0 ≤ s < +∞ : for all δ > 0, there exists a collection {Q rj (x j , t j )}

of cylinders such that E ⊆ +∞ j=1 Q rj (x j , t j ) and +∞ j=1 r s j < δ .
The relationship between the parabolic Hausdorff measure H par (E) and the classical Hausdorff measure H(E) is given by

2 H(E) -d ≤ H par (E) ≤ H(E) + 1. (2.1)
The reader is referred to [START_REF] Sweezy | The Hausdorff dimension of elliptic and elliptic-caloric measure in R N , N ≥ 3[END_REF] for further details about the parabolic framework. We remark that H par (R d+1 ) = d + 2.

2.2.

Preliminaries. First we recall an interior C 1,α regularity result for solutions of (1.3).

Proposition 2.1 ([14, Section 4.2]). If u is a viscosity solution of (1.3) in Q 1 , then u ∈ C 1,α (Q 1/2
) for some universal 0 < α < 1. Moreover, there exists some universal constant C such that sup

Q 1/2 |Du| ≤ C sup Q1 |u| + |F (0)| + g C 0,1 (Q1) .
Moreover, it is natural in the parabolic framework to introduce some other sets called parabolic balls. Precisely, given (x, t) ∈ R d+1 , we define parabolic balls of opening θ > 0 and height h ≥ 0 to be closed subsets of the following form

G θ,h (x, t) := (y, s) ∈ R d+1 : θ|y -x| 2 ≤ s -t ≤ h and G - θ,h (x, t) := (y, s) ∈ R d+1 : θ|y -x| 2 ≤ t -s ≤ h . By direct computation, |G θ,h (x, t)| = |G - θ,h (x, t)| = 2ω d d + 2 h 1+d/2 θ -d/2 , (2.2)
where ω d is the volume of the unit ball in R d . We next collect some standard material about parabolic balls. The following lemma is a slight modification of [START_REF] Wang | Small perturbation solutions for parabolic equations[END_REF]Lemma 2.2].

Lemma 2.2 (Y. Wang, [START_REF] Wang | Small perturbation solutions for parabolic equations[END_REF]). Let θ ≥ 3/4. For all (x, t) ∈ G - θ,h0 (x 0 , t 0 ) and 0 < h ≤ t 0t, there exists a cylinder Q r (x 2 , t 2 ) such that

t 2 = t + h 2 , r ≥ 1 ν h θ with ν := 4 √ 2 -1 ,
which satisfies the three following properties:

(P1 ) Q r (x 2 , t 2 ) ⊆ G θ,h (x, t) ∩ G - θ,h0 (x 0 , t 0 ) ∩ {(y, s) : t + h/4 ≤ s ≤ t + h/2}. (P2 ) |Q r (x 2 , t 2 )|/|G θ,h (x, t)| ≥ η 0 where η 0 depends only on d and θ. (P3 ) If (z, t) ∈ Q r/4 (x 2 , t 2 ), then for every (y, s) ∈ G - 1/2,r 2 /16 z, t -r 2 16 , G 1/2,t-s (y, s) ⊆ Q r (x 2 , t 2 ).
Proof. The reader is referred to [START_REF] Wang | Small perturbation solutions for parabolic equations[END_REF]Lemma 2.2] for (P1) and (P3). For (P2), we compute

|Q r (x 2 , t 2 )| = r d+2 ≥ √ 2-1 4 d+2 θ -d/2-1 h 1+d/2 = d+2 2ω d √ 2-1 4 d+2 θ -1 |G θ,h (x, t)|.
It suffices to set η 0 := d+2

2 d+3 ω d √ 2-1 4 d+2
θ -1 to get the desired estimate.

Finally, we recall a Vitali-type covering lemma for parabolic balls which was already used and proved in [START_REF] Wang | Small perturbation solutions for parabolic equations[END_REF] (and essentially follows from the standard argument for Vitali's covering lemma). It is a convenient alternative in the parabolic setting to the "stacked" estimate lemma and the Calderón-Zygmund decomposition (see [START_REF] Imbert | Introduction to fully nonlinear parabolic equations[END_REF]). Given (x, t) ∈ R d+1 and h > 0, we define the parabolic ball

G θ,h (x, t) ⊇ G θ,h (x, t) by G θ,h (x, t) := G θ,4h (x, t -3h) with θ := θ ( √ 2 + 1) 2 .
We observe from (2.2) that the following ratio is a universal constant depending only on the dimension:

η := |G θ,h (x, t)| | G θ,h (x, t)| = 4 -(1+d/2) ( √ 2 + 1) -d . (2.3)
Lemma 2.3 (Vitali's lemma for parabolic balls). Assume that E ⊆ R d+1 is bounded and h : E → R is positive. Consider the following collection of parabolic balls:

G θ,h(x,t) (x, t) : (x, t) ∈ E . If sup{h(x, t) : (x, t) ∈ E} < ∞, we can extract a countable subcollection {G θ,h(xi,ti) (x i , t i ) : i ∈ N} of disjoint parabolic balls such that E ⊆ i∈N G θ,h(xi,ti) (x i , t i ).
Finally, we give the statement of a proposition required to obtain the partial parabolic result. This proposition was obtained by Y. Wang [START_REF] Wang | Small perturbation solutions for parabolic equations[END_REF] and is the parabolic analogue of a first result of Savin [START_REF] Savin | Small perturbation solutions for elliptic equations[END_REF]. It gives C 2,α regularity for flat viscosity solutions of uniformly parabolic equations. Roughly speaking, it states that a viscosity solution of a uniformly parabolic equation that is sufficiently close to a quadratic polynomial is, in fact, a classical solution.

Proposition 2.4 (Y. Wang, [START_REF] Wang | Small perturbation solutions for parabolic equations[END_REF]). Suppose in addition to (F1)-(F2) that F (0) = 0. Suppose that 0 < α < 1 and u ∈ C(Q 1 ) is a solution of (1.1) in Q 1 . Then there exists a constant δ 0 > 0 depending only on the ellipticity constants λ and Λ, the dimension d, the modulus of continuity ω, and α, such that

sup Q1 |u| ≤ δ 0 implies that u ∈ C 2,α (Q 1/2 ).
2.3. Proof of Theorem 1.1. The strategy of the proof is the following. By a covering argument, we can cover the singular set by parabolic balls centered in points for which Ψ presents large values. This is allowed by Lemma 2.5 that shows that u is not C 2,α close to (x 0 , t 0 ) implies Ψ(x 0 , t 0 ) is large. Then the parabolic W 3,ε estimate given by Theorem 1.2 provides an upper bound on the size of the set of the bad points. Lemma 2.5. Suppose u ∈ C(Q 1 ) solves (1.1) in Q 1 and 0 < α < 1. There is a universal constant δ α > 0, depending on d, λ, Λ, ω and α such that for every (y, s 0 ) ∈ Q 1/2 0, -1 4 and 0 < r < 1/20, Ψ(u,

Q 3/4 ) ≤ r -1 δ α ∩ Q r (y, s 0 ) = ∅ implies that u ∈ C 2,α (Q r (y, s 0 -r 2 )).
Proof. Let δ > 0 to be adjusted. Suppose that 0 < r < 1/20, (y, s 0 ) ∈ Q 1/2 0, - 1 4 and (z, s) ∈ Q r (y, s 0 ) is such that

Ψ(u, Q 3/4 )(z, s) ≤ r -1 δ. Then there exist b ∈ R, p ∈ R d and M ∈ M d such that, for every (x, t) ∈ Q 3/4 such that t ≤ s, |u(x, t) -u(z, s) -p • (x -z) -b(t -s) -1 2 (x -z) • M (x -z)| ≤ 1 6 r -1 δ(|x -z| 3 + |s -t| 3/2 ). (2.4) Replacing M by 1 2 (M + M ⊤ ), we may assume that M ∈ S d . Since u is a viscosity solution of (1.1), it is clear that b + F (M ) = 0. For (x, τ ) ∈ Q 1 , define v(x, τ ) := 1 16r 2 u(z + 4rx, s + 16r 2 τ ) -u(z, s) -4rp • x -16br 2 τ -8r 2 x • M x . Noticing that (z +4rx, s+16r 2 τ ) ∈ Q 3/4 for (x, τ ) ∈ Q 1 , the inequality (2.4) implies that sup Q1 |v(x, τ )| ≤ 4 3 δ.
Define the operator F (N ) := b + F (N + M ), and observe F satisfies (F1) and (F2), with the same ellipticity constants λ, Λ and modulus ω, and

F (0) = b + F (M ) = 0. It is clear that v is a solution of ∂ t v + F (D 2 v) = 0, in Q 1 .
Let δ 0 > 0 be the universal constant in Proposition 2.4, which also depends on α. Suppose that

δ < 3 4 δ 0 .
Then Proposition 2.4 yields that v ∈ C 2,α (Q 1/2 ), from which we deduce that u ∈ C 2,α (Q 2r (z, s)). Since Q r (y, s 0r 2 ) ⊆ Q 2r (z, s), the proof is complete.

We now give the proof of the first main result.

Proof of Theorem 1.1. We assume without loss of generality that F (0) = 0. By a standard covering argument, we may fix 0 < α < 1 and assume that

Ω = Q 1 , u ∈ C(Q 1 )
is bounded, and it suffices to show that

u ∈ C 2,α (V \ Σ) with V := B 9/20 × - 1 2 , - 1 4 - 3 800 .
for a set Σ ⊆ V with H par (Σ) ≤ d + 2ε. Since, for every β > 0, the operator F β (M ) := β -1 F (βM ) satisfies both (F1) and (F2) with the same constants λ and Λ but a different modulus ω and the constant ε that we obtain does not depend on ω, we may therefore assume without loss of generality that sup Q1 |u| ≤ 1. Let Σ ⊆ V denote the set

Σ := (x, s) ∈ V : u / ∈ C 2,α Q r x, s + 1 2 r 2
for every r > 0 . Notice that Σ is closed, and thus compact. Fix 0 < r < 1/20. According to the Vitali covering theorem for parabolic cylinders [START_REF] Gary | Second order parabolic differential equations[END_REF]Lemma 7.8], there exists a finite collection {Q r (x i , s i + 1 2 r 2 )} 1≤i≤m of disjoint parabolic cylinders of radius r, with centers

(x i , s i ) ∈ Σ, such that Σ ⊆ m i=1 Q 5r x i , s i + 25 2 r 2 .
Since (x i , s i ) ∈ Σ, according to Lemma 2.5 there exists a constant δ such that

Ψ(u, Q 3/4 )(y, τ ) > r -1 δ for every (y, τ ) ∈ m i=1 Q r x i , s i + 3 2 r 2 .
Applying Theorem 1.

2 to Q r x i , s i + 3 2 r 2 ⊆ Q 1/2 0, -1 4
, we deduce that

mr d+2 = m |Q r | |Q 1 | ≤ C r -1 δ -ε
for some universal constants C, ε > 0. Therefore,

m i=1 (5r) d+2-ε ≤ 5 d+2-ε mr d+2-ε ≤ 5 d+2-ε Cδ -ε < +∞.
In particular, this implies that

H par (Σ) ≤ d + 2 -ε.
By using (2.1), we get

H(Σ) ≤ 1 2 (d + d + 2 -ε) = d + 1 - ε 2 .
3. Parabolic W 2,ε estimate

In this section, we state and prove the parabolic W 2,ε estimate associated to (1.3). It will be useful to prove Theorem 1.2 in Section 4. This result is essentially wellknown but we give the argument here for the sake of completeness.

The key argument to prove Proposition 3.1 relies on a measure estimate on small parabolic balls stated in Lemma 3.6. Its proof consists in a suitable parabolic Aleksandrov-Bakelman-Pucci (ABP) inequality and a comparison principle achieved with a barrier function. Then we derive an induction relation in Lemma 3.5 by Lemma 2.3. Finally some classical arguments permit to obtain Proposition 3.1.

Let Ω ⊆ R d+1 and u, -v ∈ LSC(Ω). We say that v touches u from below at

(x, t) ∈ Ω if v(z, τ ) ≤ u(z, τ ), for (z, τ ) ∈ Ω and τ ≤ t, v(x, t) = u(x, t).
We say that u touches v from above at (x, t)

∈ Ω if v touches u at (x, t) ∈ Ω from below. Let (y, s) ∈ R d+1 . A polynomial P is called a concave paraboloid of opening κ > 0 if P y,s;κ (z, τ ) = -κ 2 |z -y| 2 + κ(τ -s).
Similarly, a polynomial P is called a convex paraboloid of opening κ > 0 if P y,s;κ (z, τ ) = κ 2 |z -y| 2κ(τs). To state the estimate, we require some notation. Given a domain Ω ⊆ R d+1 , and a function u ∈ LSC(Ω), define the quantity

Θ(x, t) = Θ(u, Ω)(x, t) := inf A ≥ 0 : ∃p ∈ R d s.t. ∀(y, s) ∈ Ω, s ≤ t, u(y, s) ≥ u(x, t) + p • (y -x) -A 1 2 |x -y| 2 + (t -s) . Similarly, for u ∈ USC(Ω), Θ(x, t) = Θ(u, Ω)(x, t) := inf A ≥ 0 : ∃p ∈ R d s.t. ∀(y, s) ∈ Ω, s ≤ t, u(y, s) ≤ u(x, t) + p • (y -x) + A 1 2 |x -y| 2 + (t -s) , and, for u ∈ C(Ω), Θ(x, t) = Θ(u, Ω)(x, t) := max Θ(u, Ω)(x, t), Θ(u, Ω)(x, t) .
The quantity Θ(x, t) is the minimum curvature of any paraboloid that touches u from below at (x, t). If u cannot be touched from below at (x, t) by any paraboloid, then Θ(x, t) = +∞. A similar statement holds for Θ(x, t), where we touch from above instead. Moreover, a function u is C 1,1 on a closed set Γ ⊆ Ω if and only if u has tangent paraboloids from above and below with respect to Ω at each point of Γ.

The form of the W 2,ε estimate we need is given by the following proposition.

Proposition 3.1. If u ∈ LSC(Q 1 )
and L ≥ 0 satisfy the inequality

∂ t u + P + λ,Λ (D 2 u) ≥ -L in Q 1 , then for all κ > 0, |{(x, t) ∈ Q 1/2 0, -1 4 : Θ(u, Q 1 )(x, t) > κ}| ≤ C κ sup Q1 |u| + L -ε
, where the constants C and ε > 0 are universal.

We emphasize here that Θ(u, Ω) is defined in terms of quadratic polynomials that touch u at (x, t) and stay below u in the domain Ω ∩ {(y, s) : s ≤ t}, which is full in space and restricted to global times less than t.

Instead of working with the sets {Θ ≤ κκ}, we are going to consider some new sets A κ for κ > 0. We are inspired from the elliptic definition introduced by Savin [START_REF] Savin | Small perturbation solutions for elliptic equations[END_REF] and recently also used by Armstrong and Smart in [START_REF] Armstrong | Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity[END_REF]. In the parabolic setting, define, for every κ > 0,

A κ := (x, t) ∈ Q 1 : ∃(y, s) ∈ B 1 ×(-1, t] s.t. u(x, t)-inf Q1 u+κ 1 2 |x -y| 2 -(t -s) = inf (z,τ )∈Q1, τ ≤t u(z, τ ) -inf Q1 u + κ 1 2 |z -y| 2 -(τ -s) = 0 . (3.1)
It is important to notice that s takes part in the definition of A κ only to adjust the infimum to be equal to zero. Moreover, the definition of A κ given above is adapted to the domain Q 1 . It is clear how to change this definition of A κ to deal with more general domains.

The next lemma gathers some properties about the sets A κ . In particular, the link between A κ and Θ is precised. Lemma 3.2. Let u ∈ LSC(Ω), κ > 0 and A κ be defined by (3.1). Then we have

A κ ⊆ {(x, t) ∈ Q 1 : Θ(u, Q 1 )(x, t) ≤ κ} .
Moreover, for all 0 < κ 1 ≤ κ, we have

A κ1 ⊆ A κ . Proof. Let (x, t) ∈ A κ . Then (x, t) ∈ Q 1 and there exists y ∈ B 1 such that for all (z, τ ) ∈ Q 1 , τ ≤ t, we have u(z, τ ) + κ 1 2 |z -y| 2 -τ ≥ u(x, t) + κ 1 2 |x -y| 2 -t .
After rearranging the terms, we get

u(z, τ ) ≥ u(x, t) + κ 1 2 (|x -y| 2 -|z -y| 2 ) -(t -τ ) . An algebraic manipulation yields |x -y| 2 -|z -y| 2 = -|x -z| 2 + 2(y -x) • (z -x).
If we choose p = κ(yx) ∈ R d , the last inequality can be written in the form

u(z, τ ) ≥ u(x, t) + p • (z -x) -κ 1 2 (|x -z| 2 + (t -τ ) , for all (z, τ ) ∈ Q 1 , τ ≤ t, which gives precisely Θ(u, Q 1 )(x, t) ≤ κ.
For the second assertion, let (x 1 , t 1 ) ∈ A κ1 and κ > κ 1 . Hence there exist

y 1 ∈ B 1 and -1 < s 1 ≤ t 1 such that u(x 1 , t 1 ) -inf Q1 u + κ 1 1 2 |x 1 -y 1 | 2 -(t 1 -s 1 ) = inf (z,τ )∈Q1, τ ≤t u(z, τ ) -inf Q1 u + κ 1 1 2 |z -y 1 | 2 -(τ -s 1 ) = 0. (3.2)
For (y, s) ∈ R d+1 , let P be the paraboloid given by

P (z, τ ) := κ 1 2 |z -y| 2 -(τ -s) -κ 1 1 2 |z -y 1 | 2 -(τ -s 1 ) . (3.3)
Assume that we have shown there exists (y, s)

∈ B 1 × (-1, t 1 ] such that, for all (z, τ ) ∈ R d+1 , P (z, τ ) = (κ -κ 1 ) 1 2 |z -x 1 | 2 -(τ -t 1 ) . (3.4)
For this particular choice, the decomposition

u(z, τ ) -inf Q1 u + κ 1 2 |z -y| 2 -(τ -s) = u(z, τ ) -inf Q1 u + κ 1 1 2 |z -y 1 | 2 -(τ -s 1 ) + P (z, τ )
implies the result by using (3.2), P ≥ 0 on B 1 × (-1, t 1 ] and P (x 1 , t 1 ) = 0 by (3.4).

To obtain the assertion, it remains to show (3.4). By completing the square in (3.3), the polynomial P can be written in the form

P (z, τ ) = (κ -κ 1 ) 1 2 |z -z(y, s)| 2 -(τ -τ (y, s)) , with        z(y, s) := 1 κ -κ 1 (κy -κ 1 y 1 ) , τ (y, s) := 1 κ -κ 1 κs -κ 1 s 1 + 1 2 κκ 1 κ -κ 1 |y -y 1 | 2 .
We choose (y, s) to impose the condition (z(y, s), τ (y 1 , s 1 )) = (x 1 , t 1 ). This leads to select

y = 1 - κ 1 κ x 1 + κ 1 κ y 1 and s = t 1 - κ 1 κ t 1 -s 1 + 1 2 |x 1 -y 1 | 2 1 - κ 1 κ .
It is clear by convexity that y ∈ B 1 . To show that s ∈ (s

1 , t 1 ], notice t 1 -s 1 - 1 2 |x 1 -y 1 | 2 ≥ 0 by (3.
2) and use κ > κ 1 in the expression above. Now we recall the standard tool in the theory of viscosity solutions (see [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF] for further details). We denote the infimal convolution of u ∈ LSC(Q 1 ) by

u ε (x, t) = inf (z,τ )∈Q1 u(z, τ ) + 2 ε (|z -x| 2 + (τ -t) 2 ) . Moreover, if f ∈ C(Q 1 )
and

∂ t u + P + λ,Λ (D 2 u) ≥ f in Q 1 , then there exist a sequence of functions f ε ∈ C(Q 1 )
which converges locally uniformly to f respectively, as ε → 0, such that u ε satisfies

∂ t u + P + λ,Λ (D 2 u) ≥ f ε in Q 1-rε (0, T ε ), when r ε → 0 and T ε → 0 as ε → 0.
The function u ε is more regular than u and, in particular, is semiconcave. It is a good approximation to u in the sense that u ε → u locally uniformly in Q 1 as ε → 0. For us, the main utility of these approximations is the semiconcavity of u ε . If u ε can be touched from below by a smooth function ϕ at some point(y, s) ∈ Q 1 , then u ε is C 1,1 at (y, s), with norm depending only on ε and |D 2 ϕ(y, s)| and ∂ t ϕ(y, s).

The following lemma is the form of the ABP inequality we are going to use.

Lemma 3.3. Assume that L > 0 and u ∈ LSC(Q 1 ) satisfy

∂ t u + P + λ,Λ D 2 u ≥ -L in Q 1 . Suppose that a > 0 and V ⊆ R d+1 is compact such that, for each (y, s) ∈ V , there exists (x, t) ∈ Q 1 such that u(x, t) -inf Q1 u + a 2 |x -y| 2 -a(t -s) = inf (z,τ )∈Q1,τ ≤t u(z, τ ) -inf Q1 u + a 2 |z -y| 2 -a(τ -s) = 0. (3.5) Let W := {(x, t) ∈ Q 1 : (3.5) holds for u for some (y, s) ∈ V }. Then |V | ≤ 1 λ d 1 + L a + Λd d+1 |W |. (3.6)
Proof. The proof is divided into two steps.

Step 1. We make two reductions. First, by replacing u by u + α 1 2 |x| 2t and L by L + Cα and letting α → 0, we may suppose that there exists η > 0 such that for every (y, s)

∈ Q 1 , min (z,τ )∈Q1\Q1-η ,τ ≤t u(z, τ ) + a 2 |z -y| 2 -aτ > inf (z,τ )∈Q1-η,τ ≤t u(z, τ ) + a 2 |z -y| 2 -aτ . (3.7)
Next we make a reduction to the case that u is semiconcave by an infimal convolution approximation. According to (3.7), for every sufficiently small ε > 0, there exist 0 < r ε < 1 and -1 < T ε < 0 such that, for each (y, s) ∈ V , there exists

(x, t) ∈ Q 1-rε (0, T ε ) such that u ε (x, t) - inf Q1-r ε (0,Tε) u ε + a 2 |x -y| 2 -a(t -s) = inf (z,τ )∈Q1-r ε (0,Tε),τ ≤t u(z, τ ) - inf Q1-r ε (0,Tε) u ε + a 2 |z -y| 2 -a(τ -s) = 0, (3.8)
and

lim ε→0 r ε = 0 and lim ε→0 T ε = 0. Set W ε := {(x, t) ∈ Q 1-rε (0, T ε ) : (3.8) holds for u ε for some (y, s) ∈ V }.
Assume that we have shown that

lim sup ε→0 W ε ⊆ W, where lim sup ε→0 W ε := ε>0 0<δ≤ε W δ , (3.9)
and for all ε > 0,

|V | ≤ 1 λ d 1 + L a + Λd d+1 |W ε |. (3.10) Then, since sup 0<ε<1/2 |W ε | ≤ |Q 1 | < +∞, the inequality lim sup ε→0 |W ε | ≤ |lim sup ε→0 W ε |
1 λ d 1 + L a + Λd d+1 lim sup ε→0 W ε ≤ (3.9) 1 λ d 1 + L a + Λd d+1 |W | .
Thus we deduce (3.6).

To obtain the lemma, it remains to show the assertions (3.9) and (3.10). For (3.9), let (x, t) ∈ lim sup ε→0 W ε . Up to a subsequence, we can assume that

(x, t) ∈ W ε ∩ Q 1-rε (0, T ε ) for all ε > 0. By the definition of W ε , there exists (y ε , s ε ) ∈ V such that u ε (x, t) - inf Q1-r ε (0,Tε) u ε + a 2 |x -y ε | 2 -a(t -s ε ) = inf (z,τ )∈Q1-r ε (0,Tε),τ ≤t u ε (z, τ ) - inf Q1-r ε (0,Tε) u ε + a 2 |z -y ε | 2 -a(τ -s ε ) = 0. (3.11)
Since V is compact, up to extracting a subsequence, there exists (y, s)

∈ V such that (y ε , s ε ) → (y, s) as ε → 0. By convergence of u ε , we deduce that u ε (x, t) -→ ε→0 u(x, t).
Moreover, since u ε ≥ u and u ε → u locally uniformly on Q 1 , we have

lim ε→0 inf Q1-r ε (0,Tε) u ε = inf Q1 u.
Letting ε → 0 in (3.11) yields (x, t) ∈ W . This completes the proof of (3.9), and therefore it remains to prove (3.10), that is, the statement of the lemma under the extra assumption that u is semiconcave.

Step 2. Assuming u is semiconcave, we give the proof of (3.6). Select a Lebesguemeasurable function Z : V → Q 1 such that the map

(z, t) → u(z, t) -inf Q1 u + a 1 2 |z -y| 2 -(t -s)
attains its infimum in Q 1 at (z, t) = Z(y, s) and this infimum is equal to zero. For example, we may take Z(y, s) to be the lexicographically least element of the (necessarily closed) set of infima. The function

u is C 1,1 on A := Z(V ) and Z has a Lipschitz inverse Y = (y, s) given by      y(z, t) = z + 1 a Du(z, t), s(z, t) = t - 1 a u(z, t) - 1 2 |z -y(z, t)| 2 .
By Rademacher's theorem, Y is differentiable almost everywhere on A for the d+ 1dimensional Lebesgue measure. Then, by using the Lebesgue differentiation theorem, we see that u is twice differentiable in space and differentiable in time at almost every point of (z, t) ∈ A and, at such (z, t), we have

D 2 u(z, t) ≥ -aI d and ∂ t u(z, t) ≤ a.
Thus,

Dy(z, t) = I d + 1 a D 2 u(z, t) ≥ 0 as well as -λtr(Dy(z, t)) = P + λ,Λ (Dy(z, t)) = P + λ,Λ I d + 1 a D 2 u(z, t) ≥ 1 a P + λ,Λ D 2 u(z, t) + P - λ,Λ (I d ) ≥ 1 a (-L -∂ t u(z, t)) -Λd,
and therefore

0 ≤ Dy(z, t) ≤ 1 λ 1 + L a + Λd . Similarly D 2 u(z, t) ≥ -aI d implies that 1 - 1 a ∂ t u(z, t) ≤ 1 - 1 a -L -P + λ,Λ (-aI d ) = 1 + L a + Λd.
An application of the area formula for Lipschitz functions gives

|V | = ˆA | det DY (z, t)|dzdt = ˆA det I + 1 a D 2 u(z, t) 1 - 1 a ∂ t u(z, t) dzdt ≤ 1 λ d 1 + L a + Λd d 1 + L a + Λd |A|,
from which we obtain the lemma, using that A ⊆ W .

In our analysis, an important role will be played by the functions φ which we define by

φ(x, t) := c(t + τ ) -b e -a |x| 2 t+τ -e -aθ -1 .
The parameters a, b and c will be adjusted with the uniform ellipticity constants and the opening θ of the parabolic balls of the form G θ,1+τ (0, -τ ) with τ > 0.

More precisely, we will consider the choice given by

a = max 1 + dΛθ 2λ , θ (3.12
         ∂ t φ + P + λ,Λ (D 2 φ) ≤ -1, on G θ,1+τ (0, -τ ) ∩ {(y, s) : s > 0}, φ = 0, on ∂ p G θ,1+τ (0, -τ ) ∩ {(y, s) : s > 0}, φ > 0, on G θ,1+τ (0, -τ ) \ ∂ p G θ,1+τ (0, -τ ), 0 ≤ φ ≤ β, on G θ,1+τ (0, -τ ) ∩ {(y, s) : s = 0},
with β > 0 given by

β := 2 (1 + τ ) b+1 τ b (e aθ -1 -1).
Proof. Let us introduce the variable ρ := |x| 2 /(t + τ ) and the function ψ given by ψ(ρ, t) := c(t + τ ) -(b+1) e -aρ .

First, by inserting the value of c given by (3.14), observe that

ψ(ρ, t) ≥ 2 on G θ,1+τ (0, -τ ). (3.15)
The two last properties are immediate to check. Thus we focus on the first assertion. The time derivative of φ is given by

∂ t φ(x, t) = ψ(ρ, t) -(1 -e a(ρ-θ -1 ) )b + aρ .
and the Hessian of φ is given by

D 2 φ(x, t) = 2aψ(ρ, t) (-1 + 2aρ) x ⊗ x |x| 2 -I - x ⊗ x |x| 2 .
and has eigenvalues 2aψ(ρ, t) (-1 + 2aρ) with multiplicity 1 and -2aψ(ρ, t) with multiplicity d -1. Hence

P + λ,Λ (D 2 φ(x, t)) = ψ(ρ, t) 2a (d -2aρ) Λ, if ρ ≤ (2a) -1 , 2a (-λ (-1 + 2aρ) + (d -1)Λ) , if ρ ≥ (2a) -1 . We distinguish two cases. If 0 ≤ ρ ≤ (2a) -1 , ∂ t φ(x, t) + P + λ,Λ (D 2 φ(x, t)) = ψ(ρ, t) -(1 -e a(ρ-θ -1 ) )b + a(ρ + 2(d -2aρ)Λ) ≤ (3.12) ψ(ρ, t) -(1 -e 1 2 -aθ -1 )b + 1 2 + 2daΛ ≤ (3.13)
-1 2 ψ(ρ, t).

By using (3.15), we obtain the desired upper bound. Now assume that (2a

) -1 ≤ ρ ≤ θ -1 , ∂ t φ(x, t) + P + λ,Λ (D 2 φ(x, t)) ≤ ψ(ρ, t) -(1 -e a(ρ-θ -1 ) )b + a(1 -4aλ)ρ + 2daΛ ≤ ψ(ρ, t) -b + 2adΛ + be a(ρ-θ -1 ) + a(1 -4aλ)ρ .
The function ρ → be a(ρ-θ -1 ) + a(1 -4aλ)ρ is decreasing on ] -∞, ρ 0 ] and increasing on [ρ 0 , +∞[ with ρ 0 := 1 a ln 4aλ-1 b + θ -1 . By (3.13), the coefficients a and b are chosen so that ρ 0 < (2a) -1 . Under this assumption, the upper bound on the interval

[(2a) -1 , θ -1 ] corresponds to ρ = θ -1 which provides ∂ t φ(x, t) + P + λ,Λ (D 2 φ(x, t)) ≤ ψ(ρ, t)a 2dΛ + θ -1 -4θ -1 aλ ≤ (3.12) -ψ(ρ, t)aθ -1 ≤ (3.12)
-ψ(ρ, t).

By recalling (3.15), we obtain the desired upper bound.

The following lemma contains the measure theoretic information necessary to conclude the proof of Proposition 3.1. The argument relies on Lemmas 3.3 and 3.4. Lemma 3.5. Let 3/4 ≤ θ, h 0 > 0, κ 1 > 0, A κ be defined by (3.1) and G - θ,h0 (x 0 , t 0 ) ⊆ Q 1 . There exist constants M ≥ 1 and σ > 0 depending only on θ and d, λ, Λ such that, if (x 0 , t 0 ) ∈ A κ1 and κ ≥ κ 1 , then

|A Mκ ∩ G - θ,h0 (x 0 , t 0 )| ≥ |G - θ,h0 (x 0 , t 0 ) ∩ A κ | + ση|G - θ,h0 (x 0 , t 0 ) \ A κ |. Proof.
We decompose the measure estimate into two parts

|A Mκ ∩ G - θ,h0 (x 0 , t 0 )| = |A κ ∩ G - θ,h0 (x 0 , t 0 )| + |(A Mκ \ A κ ) ∩ G - θ,h0 (x 0 , t 0 )|. It is enough to estimate |(A Mκ \ A κ ) ∩ G - θ,h0 (x 0 , t 0 )|. We claim that G - θ,h0 (x 0 , t 0 ) \ A κ ≤ 1 ση (A Mκ \ A κ ) ∩ G - θ,h0 (x 0 , t 0 ) .
For κ ≥ κ 1 , we define the collection of parabolic balls given by

B := G θ,h (x, t) : (x, t) ∈ G - θ,h0 (x 0 , t 0 ), G θ,h (x, t) ∩ G - θ,h0 (x 0 , t 0 ) ∩ {(y, s) : s < t + h} ⊆ Q 1 \ A κ and G θ,h (x, t) ∩ {(y, s) : s = t + h} ∩ G - θ,h0 (x 0 , t 0 ) ∩ A κ = ∅ .
Notice that for all (x, t) ∈ G - θ,h0 (x 0 , t 0 ), the point (x 0 , t 0 ) belongs to the parabolic ball G θ,t0-t (x, t). Observe that (x 0 , t 0 ) ∈ A κ by applying Lemma 3.2 with (x 0 , t 0 ) ∈ A κ1 . This implies that for all G θ,h (x, t) ∈ B, h ≤ t 0t ≤ h 0 . Then, by Lemma 2.3, we may extract from B a countable subcollection {G θ,hi (x i , t i ) : i ∈ N} such that the G θ,hi (x i , t i ) are disjoint,

G - θ,h0 (x 0 , t 0 ) \ A κ ⊆ i∈N G θ,hi (x i , t i ) and |G θ,hi (x i , t i )| | G θ,hi (x i , t i )| = η,
with η given by (2.3). By combining these, we get

|G - θ,h0 (x 0 , t 0 ) \ A κ | ≤ i∈N | G θ,hi (x i , t i )| = 1 η i∈N |G θ,hi (x i , t i )|.
Next we complete the proof under the assumption that for all i ∈ N,

|G θ,hi (x i , t i ) ∩ G - θ,h0 (x 0 , t 0 ) ∩ A Mκ | ≥ σ|G θ,hi (x i , t i )| (3.16)
for some constants M > 1 and σ > 0, depending only on θ, d, λ and Λ. Using also that the selected balls are disjoint, we obtain that

i∈N |G θ,hi (x i , t i )| ≤ 1 σ i∈N G θ,hi (x i , t i ) ∩ G - θ,h0 (x 0 , t 0 ) ∩ A Mκ = 1 σ i∈N G θ,hi (x i , t i ) ∩ {(y, s) : s < t i + h i } ∩ G - θ,h0 (x 0 , t 0 ) ∩ A Mκ . Since every ball in B satisfies G θ,h (x, t) ∩ {(y, s) : s < t + h} ∩ G - θ,h0 (x 0 , t 0 ) ⊆ G - θ,h0 (x 0 , t 0 ) \ A κ , we deduce that |G - θ,h0 (x 0 , t 0 ) \ A κ | ≤ 1 ση G - θ,h0 (x 0 , t 0 ) ∩ (A Mκ \ A κ ) .
The proof is complete, pending the verification of (3.16), which is achieved in the next lemma.

The following lemma is the key step in the proof of Proposition 3.1.

Lemma 3.6. Let 3/4 ≤ θ, κ 1 > 0 and G - θ,h0 (x 0 , t 0 ) ⊆ Q 1 . Suppose u ∈ LSC(Q 1 ) satisfies ∂ t u + P + λ,Λ (D 2 u) ≥ 0. (3.17)
There exist constants M ≥ 1 and σ > 0 depending only on θ and d, λ, Λ such that if

(x 0 , t 0 ) ∈ A κ , κ ≥ κ 1 , then for all (x, t) ∈ G - θ,h0 (x 0 , t 0 ) satisfying G θ,h (x, t) ∩ {(y, s) : s = t + h} ∩ G - θ,h0 (x 0 , t 0 ) ∩ A κ = ∅, we have |G θ,h (x, t) ∩ A Mκ ∩ G - θ,h0 (x 0 , t 0 )| ≥ σ|G θ,h (x, t)|. Proof. Let (z 1 , t 1 ) ∈ G θ,h (x, t) ∩ {(y, s) : s = t + h} ∩ G - θ,h0 (x 0 , t 0 ) ∩ A κ .
By the definition of A κ , we can touch u at (z 1 , t 1 ) from below by a concave paraboloid of opening κ: there exist

y 1 ∈ B 1 and -1 < s 1 ≤ t 1 such that u(z 1 , t 1 ) -inf Q1 u + κ 2 |z 1 -y 1 | 2 -κ(t 1 -s 1 ) = inf (z,τ )∈Q1,τ ≤t1 u(z, τ ) -inf Q1 u + κ 2 |z -y 1 | 2 -κ(τ -s 1 ) = 0. (3.18)
Step 1. Let Q r (x 2 , t 2 ) ⊆ G θ,h (x, t)∩G - θ,h0 (x 0 , t 0 ) be the cylinder given by Lemma 2.2 (see Figure 1). In particular,

t 2 = t + h 2 and r ≥ 1 ν h θ with ν = 4 √ 2 -1 . (3.19)
If we set

α := min 1, θ (1 + √ 2) 2 and δ := 1 16ν 2 α θ > 0, (3.20)
we claim that the parabolic ball (see Figure 1)

G 2 := G α,h( 1 2 +δ) (x 2 , t 2 -δh) satisfies G 2 ∩ {(y, s) : s ≤ t 2 } ⊆ Q r/4 (x 2 , t 2 ) (3.21) and (z 1 , t 1 ) ∈ (G 2 \ ∂ p G 2 ) ∩ {(y, s) : s = t 1 }. ( 3
) ∈ G 2 satisfies |z -x 2 | 2 ≤ α -1 δh ≤ r 2 16
. Then, for the second assertion, observing that (z

1 , t 1 ) ∈ G θ,h (x, t) ∩ {(y, s) : s = t + h}, it is enough to show that G θ,h (x, t) ∩ {(y, s) : s = t + h} ⊆ (G 2 \ ∂ p G 2 ) ∩ (y, s) : s = t 2 + h 2 .
Let (z, τ ) ∈ G θ,h (x, t) ∩ {(y, s) : s = t + h}. Since (x 2 , t 2 ) ∈ G θ,h (x, t) ∩ {(y, s) : s = t + h/2} and δ > 0, we get

|z -x 2 | ≤ |z -x| + |x -x 2 | = 1 + 1 √ 2 h θ < (3.20) 1 2 + δ h α . This is equivalent to (z, τ ) ∈ (G 2 \ ∂ p G 2 ) ∩ (y, s) : s = t 2 + h 2 , which gives (3.22).
Step 2. We claim that there exists (z

2 , t 2 ) ∈ G 2 ∩ {(y, s) : s = t 2 } such that u(z 2 , t 2 ) -inf Q1 u + κ 2 |z 2 -y 1 | 2 -κ(t 2 -s 1 ) ≤ (dΛ + 3)κβθν 2 r 2 . (3.23)
By applying (a properly scaled) Lemma 3.4, there exists a barrier function w which satisfies

     ∂ t w + P + λ,Λ (D 2 w) ≤ -1, in G 2 ∩ {(y, s) : s > t 2 }, w = 0, on ∂ p G 2 ∩ {(y, s) : s > t 2 }, 0 ≤ w ≤ βh, on G 2 ∩ {(y, s) : s = t 2 }, (3.24)
and w > 0 in (G 2 \ ∂ p G 2 ) ∩ {(y, s) : s > t 2 }. In particular, this implies by Step 1 that w(z 1 , t 1 ) > 0. We have that w ≤ βh in G 2 ∩ {(y, s) : s > t 2 } by the maximum principle. Observe that the function

ϕ(z, τ ) := ((dΛ + 1)κ + 2κ 1 )w -κ 2 |z -y 1 | 2 + κτ, (z 1 , t 1 ) (x 2 , t 2 ) h h 2 (x, t) G 2 G θ,h (x, t) (x 0 , t 0 ) G - θ,h0 (x 0 , t 0 ) Qr (x 2 , t 2 ) r/4 r Q r (x 2 , t 2 ) Q r/4 (x 2 , t 2 ) t 2 -δh (x 2 , t 2 ) G 2 Figure 1. Q r (x 2 , t 2 ) ⊆ G θ,h (x, t) ∩ G - θ,h0 (x 0 , t 0 ) and G 2 = G α,h(1/2+δ) (x 2 , t 2 -δh). satisfies ∂ t ϕ + P + λ,Λ (D 2 ϕ) ≤ -2κ 1 . Notice that u satisfies ∂ t u + P + λ,Λ (D 2 u) ≥ -κ 1 . (3.25)
The comparison principle implies that the map (z, τ

) → u(z, τ ) -ϕ(z, τ ) attains its infimum in G 2 ∩ {(y, s) : s > t 2 } at some point (z, τ ) = (z 2 , t 2 ) ∈ ∂ p (G 2 ∩ {(y, s) : s > t 2 }). It is impossible that (z 2 , t 2 ) ∈ ∂ p G 2 ∩ {(y, s) : s > t 2 } since (3.18), w = 0 on ∂ p G 2 ∩ {(y, s) : s > t 2 } and w(z 1 , t 1 ) > 0 imply that u(z 1 , t 1 )-ϕ(z 1 , t 1 ) = u(z 1 , t 1 ) + κ 2 |z 1 -y 1 | 2 -κt 1 -((dΛ + 1)κ + 2κ 1 )w(z 1 , t 1 ) < (3.18) inf (z,τ )∈Q1 τ ≤τ1 u(z, τ ) + κ 2 |z -y 1 | 2 -κτ ≤ inf (z,τ )∈∂pG2 t2<τ ≤t1 u(z, τ ) + κ 2 |z -y 1 | 2 -κτ = inf (z,τ )∈∂pG2 t2<τ ≤t1 (u(z, τ ) -ϕ(z, τ )) .
Moreover, it is impossible that (z 2 , t 2 ) satisfies t 2 = t 1 since ϕ satisfies

∂ t ϕ(•, t 1 ) + P + λ,Λ (D 2 ϕ(•, t 1 )) ≤ -2κ 1 for G 2 ∩ {(y, s) : s = t 1 }. Hence (z 2 , t 2 ) ∈ G 2 ∩ {(y, s) : s = t 2 }
and so, in particular, by (3.21) satisfies

|z 2 -x 2 | ≤ 1 4 r, and ϕ(z 2 , t 2 ) = ((dΛ + 1)κ + 2κ 1 )w(z 2 , t 2 ) -κ 2 |z 2 -y 1 | 2 + κt 2 . Using that w > 0 in G 2 ∩ {(y, s) : s > t 2 }, we obtain that u(z 1 , t 1 ) + κ 2 |z 1 -y 1 | 2 -κt 1 ≥ u(z 1 , t 1 ) -ϕ(z 1 , t 1 ) ≥ inf G2∩{s>t2} (u(z, τ ) -ϕ(z, τ )) = u(z 2 , t 2 ) -ϕ(z 2 , t 2 ) = u(z 2 , t 2 ) + κ 2 |z 2 -y 1 | 2 -κt 2 -((dΛ + 1)κ + 2κ 1 )w(z 2 , t 2 )
. By combining (3.19) and (3.24), we know that

w(z 2 , t 2 ) ≤ βθν 2 r 2 .
Using this together with (3.18), (z Step 3. Let γ := 17(dΛ+3)βθν 2 . We claim that for all (y 2 , s 2 ) ∈ G - 1/2,r 2 /16 z 2 , t 2 -r 2 16 , there exists a point (z

1 , t 1 ) ∈ (G 2 \ ∂ p G 2 ) ∩ {(y, s) : s = t 2 + h 2 } (by (3.22)) and κ ≥ κ 1 , inf (z,τ )∈Q1,τ ≤t1 u(z, τ ) + κ 2 |z -y 1 | 2 -κτ ≥ u(z 2 , t 2 ) + κ 2 |z 2 -y 1 | 2 -κt 2 -(dΛ + 3) κβθν 2
3 , t 3 ) ∈ G 1/2,t2-s2 (y 2 , s 2 ) ⊆ Q r (x 2 , t 2 ) such that χ(z 3 , t 3 ) = inf (z,τ )∈Q1 τ ≤t3 χ(z, τ ) = 0, (3.26)
where the function χ is given by

χ(z, τ ) := u(z, τ ) -inf Q1 u + κ 2 |z -y 1 | 2 -κ(τ -s 1 ) + γκ 2 |z -y 2 | 2 -γκ(τ -s 2 ).
Observe the function χ can be written in the form

χ(z, τ ) = u(z, τ ) -inf Q1 u -P y1,s1;κ (z, τ ) -P y2,s2;γκ (z, τ )
where P y1,s1;κ and P y2,s2;γκ are the concave paraboloids

P y1,s1;κ (z, τ ) := -κ 2 |z -y 1 | 2 + κ(τ -s 1 ) P y2,s2;γκ (z, τ ) := -γκ 2 |z -y 2 | 2 + γκ(τ -s 2 ). Let (y 2 , s 2 ) ∈ G - 1/2,r 2 /16 z 2 , t 2 -r 2 16
. By combining Steps 1-2, observe that

(z 2 , t 2 ) ∈ G 2 ∩ {(y, s) : s = t 2 } ⊆ Q r/4 (x 2 , t 2 ).
By applying Lemma 2.2 part (P3), we get G 1/2,t2-s2 (y 2 , s 2 ) ⊆ Q r (x 2 , t 2 ). Define

ζ(t) := inf χ(z, τ ) : (z, τ ) ∈ G 1/2,t2-s2 (y 2 , s 2 ), τ ≤ t .
To prove the claim, we will use the three following facts:

t → inf χ(z, τ ) : (z, τ ) ∈ Q 1 \ G 1/2,
: (z, τ ) ∈ Q 1 , τ ≤ t} = min ζ(t), inf χ(z, τ ) : (z, τ ) ∈ Q 1 \ G 1/2,
(z, τ ) ∈ Q 1 \ G 1/2,t2-s2 (y 2 , s 2 ), τ ≤ t 2 , τ -s 2 -|z-y2| 2 2 < 0,
and so,

χ(z, τ ) > u(z, τ ) -inf Q1 u -P y1,s1;κ (z, τ ).
By (3.18), the right-hand side in the inequality above is nonnegative. Therefore, we have χ(z, τ ) > 0 for all (z, τ

) ∈ Q 1 \ G 1/2,t2-s2 (y 2 , s 2 ), τ ≤ t 2 ≤ t 1 .
By passing to the inf, we obtain (3.27). By repeating (3.18), we also obtain that

χ(y 2 , s 2 ) = u(y 2 , s 2 ) -inf Q1 u -P y1,s1;κ (y 2 , s 2 ) ≥ 0. For (3.28), since (y 2 , s 2 ) ∈ G - 1/2,r 2 /16 z 2 , t 2 -r 2 16
, we have

t 2 -s 2 -|y2-z2| 2 2 ≥ r 2
16 . This implies P y2,s2;γκ (z 2 , t 2 ) ≥ γκ r 2 16 . By using also Step 2, we obtain χ(z 2 , t 2 ) < (dΛ + 3)κβθν 2 r 2γκ r 2 16 . By inserting the value of γ, we get (dΛ + 3)κβθν 2 r 2γκ r 2 16 = κr 2 16 16(dΛ + 3)βθν 2γ < 0, and the claim (3.28) follows.

For (3.29), it is clear that ζ is non-increasing. Moreover, ζ ∈ LSC(Q 1 ) since u ∈ LSC(Q 1 ). The lower semicontinuity and the monotonicity of ζ imply that ζ is right-continuous. To show that ζ is left-continuous, we argue by contradiction. Assume there exist t ∈ (s 2 , t 2 ] and a strictly increasing sequence r k → t such that

ζ(r k ) -→ k→+∞ ζ -(t) > ζ(t), with ζ(t) < 0, (3.30)
where ζ -(τ ) denotes the limit from the left of ζ at τ . Define P (z, τ ) := P y1,s1;κ (z, τ ) + P y2,s2;γκ (z, τ ).

By (3.30), we deduce that for all (z, τ

) ∈ G 1/2,t2-s2 (y 2 , s 2 ), τ < t, u(z, τ ) -inf Q1 u -P (z, τ ) ≥ ζ -(t), (3.31)
and there exists (z, t) ∈ G 1/2,t2-s2 (y 2 , s 2 ) such that

ζ(t) = u(z, t) -inf Q1 u -P (z, t) = χ(z, t) = inf χ(z, τ ) : (z, τ ) ∈ G 1/2,t2-s2 (y 2 , s 2 ), τ ≤ t . (3.32)
Notice that by using (3.18), ζ(t) < 0 implies that (z, t) ∈ G 1/2,t2-s2 (y 2 , s 2 ) \ ∂ p G 1/2,t2-s2 (y 2 , s 2 ). Let φ be a smooth function such that uφ has a local minimum at (z, t). Denote φ the map φ(y, s) := φ(y, s) -L(st) with L ≥ 0 to be selected below. For s ≥ t, using that φ(z, t) = φ(z, t), we obtain On the set

u(y, s) -φ(y, s) = (u -φ)(y, s) + L(s -t) ≥ (u -φ)(z, t). ( 3
(y, s) ∈ G 1/2,t2-s2 (y 2 , s 2 ) \ ∂ p G 1/2,t2-s2 (y 2 , s 2 ) : s ≤ t and max |P (y, s) -P (z, t)|, |φ(y, s) -φ(z, t)|, L|s -t| ≤ 1 4 (ζ -(t) -ζ(t)) ,
the following inequality holds true

u(y, s) -φ(y, s) > u(z, t) -φ(z, t) + 1 4 (ζ -(t) -ζ(t)), (3.34)
and by putting together the two cases (3.33) and (3.34), the map (y, s) → (uφ)(y, s) has a local minimum at (z, t). Using that u is a supersolution of (3.17), we obtain

0 ≤ ∂ t φ(z, t) + P + λ,Λ (D 2 φ(z, t)) = -L + ∂ t φ(z, t) + P + λ,Λ (D 2 φ(z, t)).
Taking L := 1 + ∂ t φ(z, t) + P + λ,Λ (D 2 φ(z, t)) ≥ 1, we get a contradiction.

Step 4. Consider the function Z : G - 1/2,r 2 /16 z 2 , t 2 -r 2 16 → Q 1 given by Z(y, s) = (z(y, s), τ (y, s)), where

   z(y, s) := 1 γ+1 (y 1 + γy), τ (y, s) := 1 γ+1 s 1 + γs -γ (γ+1) 2 |y 1 -y| 2 .
Observe by completing the square that we have for all

z ∈ R d , τ ∈ R, κ 2 |z -y 1 | 2 -κ(τ -s 1 ) + γκ 2 |z -y 2 | 2 -γκ(τ -s 2 ) = (γ+1)κ 2 |z -z(y 2 , s 2 )| 2 -(γ + 1)κ(τ -τ (y 2 , s 2 )).

It follows by

Step 3 that the map

(z, τ ) → u(z, τ ) -inf Q1 u + (γ+1) 2 |z -z(y 2 , s 2 )| 2 -(γ + 1)(τ -τ (y 2 , s 2 ))
attains its infimum in Q 1 at some point of Q r (x 2 , t 2 ) and this infimum is equal to zero. Since u satisfies (3.25), we can apply Lemma 3.3 by taking L = κ 1 > 0 and we obtain

Z G - 1/2,r 2 /16 z 2 , t 2 -r 2 16 ≤ 1 λ d 1 + κ 1 (γ + 1)κ + Λd d+1 |Q r (x 2 , t 2 ) ∩ A (γ+1)κ |. (3.35)
Since γ ≥ 1 and γ/(γ + 1) ≥ 1 2 , we deduce by the change of variables formula that

Z G - 1/2,r 2 /16 z 2 , t 2 -r 2 16 ≥ 2 -d-1 G - 1/2,r 2 /16 z 2 , t 2 -r 2 16 . (3.36)
By combining the explicit expression of a parabolic ball and (2.2), we deduce there exists a numerical constant 0 < c < 1 such that

G - 1/2,r 2 /16 z 2 , t 2 -r 2 16 ≥ c θ |G θ,h (x, t)|. (3.37) By Lemma 2.2 part (P1), Q r (x 2 , t 2 ) ⊆ G θ,h (x, t) ∩ G - θ,h0 (x 0 , t 0 )
. By combining this observation with (3.35), (3.36) and (3.37), recalling that κ ≥ κ 1 and taking M := γ + 1, we obtain that

|G θ,h (x, t) ∩ A Mκ ∩ G - θ,h0 (x 0 , t 0 )| ≥ 2 -d-1 λ d (1 + 1 γ+1 + Λd) d+1 c θ :=σ |G θ,h (x, t)|,
as desired.

We next present the proof of the parabolic W 2,ε estimate.

Proof of Proposition 3.1. We begin with four reductions. First, we may assume that L = 0. Otherwise we replace u by u := u + Lt, which is solution of

∂ t u + P + λ,Λ (D 2 u) ≥ 0. Let (x, t) ∈ Q 1 and A > Θ( u, Q 1 )(x, t). By the definition of Θ( u, Q 1 )(x, t), u(y, s) + Ls ≥ u(x, t) + Lt + p • (x -y) -A 1 2 |x -y| 2 + (t -s) .
Since L ≥ 0 and ts ≥ 0, we deduce that, for all (y, s)

∈ Q 1 , s ≤ t, u(y, s) ≥ u(x, t) + p • (x -y) -A 1 2 |x -y| 2 + (t -s) .
Then A ≥ Θ(u, Q 1 )(x, t), and so Θ( u, Q 1 )(x, t) ≥ Θ(u, Q 1 )(x, t). Under the assumption the estimate holds true for u, we get

|{(x, t) ∈ Q 1/2 0, -1 4 : Θ(u,Q 1 )(x, t) > κ}| ≤ |{(x, t) ∈ Q 1/2 0, -1 4 : Θ( u, Q 1 )(x, t) > κ}| ≤ C κ sup Q1 | u| -ε .
Observing that sup Q1 | u| ≤ sup Q1 |u| + L, we get the desired estimate for u.

Next, using the positive homogeneity of P + λ,Λ and Θ and replacing u by u := u/ sup Q1 |u|, we may assume that sup Q1 |u| = 1. Finally, by Lemma 3.2, we have

|{(x, s) ∈ Q 1/2 0, -1 4 : Θ(u, Q 1 )(x, s) > κ}| ≤ Q 1/2 0, -1 4 \ A κ .
Thus, it suffices to prove that there exist some universal constants C, κ 0 , ε > 0 such that for all κ ≥ κ 0 ,

Q 1/2 0, -1 4 \ A κ ≤ C κ κ 0 -ε . (3.38)
Step 1 (Geometric configuration). Fix 3 4 ≤ θ ≤ 5. We consider the cube

C R (0, T 0 ) := 0 + (-R, R) d × (T 0 -H cube , T 0 ], with T 0 ∈ (-1 2 , -1 4 ], R > 0 which satisfies R < 1 (2 + 16θ) √ d , (3.39)
and H cube , depending on R and θ, given by

H cube := (1 + 5θ)dR 2 .
To perform our analysis, we introduce two parabolic balls G 1/2,H1 (0, T 1 ) and G 1/2,H2 (0, T 2 ) (see Figure 2) where the parameters T 1 , H 1 , T 2 and H 2 denote

H 1 := 1 2 dR 2 and T 1 := T 0 + 4dθR 2 , H 2 := 1 8 dR 2 and T 2 := T 1 + H 1 -H 2 = T 1 + 3 8 dR 2
, and a parabolic ball G - θ,H0 (x 0 , t 0 ) with (x 0 , t 0 ) selected at Step 2 and

H 0 := T 1 -T 0 + H 1 + H cube = 3 2 + 9θ dR 2 .
To be well-defined, our argument requires that G 1/2,H1 (0, T 1 ) and G 1/2,H2 (0, T 2 ) are contained in Q 1 . In particular, we need to check the condition T 1 + H 1 < 0. By using the explicit expressions of T 1 , H 1 and (3.39), we get

T 1 + H 1 = T 0 + 4dθR 2 + 1 2 dR 2 ≤ -1 4 + 1 2 + 4θ dR 2 < 0.
Step 2 (Existence of the paraboloid for a certain κ = κ 0 ). We claim that there exists (x 0 , t 0 ) ∈ G 1/2,H1 (0, T 1 ) such that there exists (y 0 , s 0

) ∈ B 1 × (-1, t 0 ] such that u(x 0 , t 0 ) -inf Q1 u + κ 2 |x 0 -y 0 | 2 -κ(t 0 -s 0 ) = inf (z,τ )∈Q1,τ ≤t0 u(z, τ ) -inf Q1 u + κ 2 |z -y 0 | 2 -κ(τ -s 0 ) = 0. (3.40) In addition, C R (0, T 0 ) ⊆ G - θ,H0 (x 0 , t 0 ) ⊆ Q 1 , (3.41)
and there exists 0 < ξ < 1, depending only on θ, such that

|C R (0, T 0 )| ≥ ξ|G - θ,H0 (x 0 , t 0 )|. (3.42)
To prove the claim given by (3.40), first we are going to find (x 0 , t 0 ) realizing the infimum for a good choice of y 0 and κ chosen sufficiently large. Then we are going to determine s 0 such that this infimum will be equal to zero and we are going to check that -1 < s 0 ≤ t 0 to complete the proof. First select (y 0 , τ 0 ) ∈ G 1/2,H2 (0, T 2 ) such that

u(y 0 , τ 0 ) = inf u(z, τ ) : (z, τ ) ∈ G 1/2,H2 (0, T 2 ) , H 1 H 2 (y 0 , τ 0 ) (x 0 , t 0 ) (0, T2) G 1/2,H2 (0, T 2 ) (0, T1) G 1/2,H1 (0, T 1 ) (0, T0) H Ù R C R (0, T 0 ) H 0 Figure 2.
Geometry in the proof of Proposition 3.1: the cube C R (0, T 0 ) and the parabolic balls G 1/2,H1 (0, T 1 ) and

G 1/2,H2 (0, T 2 ) such that C R (0, T 0 ) ⊆ G - θ,H0 (x 0 , t 0 ) with (x 0 , t 0 ) ∈ G 1/2,H1 (0, T 1 ).
and κ such that κ ≥ κ 0 with κ 0 := max 24, 320 dR 2 .

(3.43)

Then for all (z, τ

) ∈ Q 1 , u(z, τ ) + κ 2 |z -y 0 | 2 -κτ ≥ u(y 0 , τ 0 ) -κτ 0 -osc Q1 u + κ 1 2 |z -y 0 | 2 -(τ -τ 0 ) . We check that T 2 -T 1 > 2H 2 . For all (z, τ ) ∈ Q 1 \ G 1/2,H1 (0, T 1 ), τ ≤ τ 0 , we have 1 2 |z -y 0 | 2 -(τ -τ 0 ) ≥ 1 2 √ T 2 -T 1 - √ 2H 2 2 , if τ ∈ 1 2 (T 1 + T 2 ) , τ 0 , -1 2 (T 1 + T 2 ) + τ 0 , if τ < 1 2 (T 1 + T 2 ) .
By inserting the values of T 1 , T 2 and by using that τ 0 ∈ G 1/2,H2 (0, T 2 ), we get

1 2 |z -y 0 | 2 -(τ -τ 0 ) ≥ 1 160 dR 2 , if τ ∈ 1 2 (T 1 + T 2 ) , τ 0 , 3 16 dR 2 , if τ < 1 2 (T 1 + T 2 ) .
Since osc Q1 u ≤ 2, and recalling (3.43), this implies that

inf u(z, τ ) + κ 2 |z -y 0 | 2 -κτ : (z, τ ) ∈ Q 1 \ G 1/2,H1 (0, T 1 ), τ ≤ τ 0 ≥ u(y 0 , τ 0 ) -κτ 0 .
Thus there exists (x 0 , t 0 ) ∈ G 1/2,H1 (0, T 1 ), t 0 ≤ τ 0 , such that

u(x 0 , t 0 ) + κ 2 |x 0 -y 0 | 2 -κt 0 = inf (z,τ )∈Q1,τ ≤τ0 u(z, τ ) + κ 2 |z -y 0 | 2 -κτ = inf (z,τ )∈Q1,τ ≤t0 u(z, τ ) + κ 2 |z -y 0 | 2 -κτ .
Now let s 0 ≤ t 0 be defined by

s 0 := t 0 - 1 κ u(x 0 , t 0 ) -inf Q1 u - 1 2 |x 1 -y 0 | 2 .
Thus, the infimum in (3.40) is equal to zero. To complete the proof of the claim, it remains to check that s 0 > -1. By using that u(x 0 , t 0 )inf Q1 u ≤ 2, we get

t 0 -s 0 ≤ 2 κ + 1 2 |x 0 -y 0 | 2 .
Since (x 0 , t 0 ), (y 0 , τ 0 ) ∈ G θ,H1 (0, T 1 ), we get by inserting (3.39) and using θ ≥ 3/4 that

|x 0 -y 0 | 2 ≤ 2H 1 θ = dR 2 θ ≤ 1 4θ(1 + 8θ) 2 ≤ 1 147 ≤ 1 6 .
Thus we obtain

t 0 -s 0 ≤ 2 κ + 1 12 .
By using (3.43), we get

t 0 -1 6 ≤ s 0 ≤ t 0 . Since t 0 ∈ [T 1 , T 1 + H 1 ] = [T 0 + 4dθR 2 , T 0 + 4θ + 1
2 )dR 2 , in particular, -1 2 ≤ T 0 ≤ t 0 ≤ 0 and we conclude that s 0 > -1. This completes the proof of (3.40).

To show (3.41), first notice that |x 0 | ≤ √ dR since (x 0 , t 0 ) ∈ G 1/2,H1 (0, T 1 ). Now observe that each (y, s) ∈ C R (0, T 0 ) satisfies t 0s ≤ H 0 and

|y -x 0 | ≤ |y| + |x 0 | ≤ √ dR + √ dR = 2 √ dR.
This means (y, s) ∈ G - θ,H0 (x 0 , t 0 ) since T 1 -T 0 ≥ 4dθR 2 (see also Figure 2). By a direct computation, we check (3.42).

Step 3. By Step 2, the point (x 0 , t 0 ) belongs to A κ0 . By Lemma 3.2, we deduce that, for all κ ≥ κ 0 , (x 0 , t 0 ) belongs to A κ . Then we can apply Lemma 3.5 and we get that for all κ ≥ κ 0 ,

|A Mκ ∩ G - θ,H0 (x 0 , t 0 )| ≥ |G - θ,H0 (x 0 , t 0 ) ∩ A κ | + ση|G - θ,H0 (x 0 , t 0 ) \ A κ |.
After rearranging the terms, this implies

|G - θ,H0 (x 0 , t 0 ) \ A Mκ | ≤ (1 -ση)|G - θ,H0 (x 0 , t 0 ) \ A κ |. (3.44)
Step 4. We claim that for all κ > κ 0 , we have

|C R (0, T 0 ) \ A κ | ≤ 1 ξ(1 -ση) κ κ 0 -ε |C R (0, T 0 )|,
where ε is given by ε := -ln (1-ση) ln M > 0. First we obtain the decay measure estimate on the parabolic ball G - θ,H0 (x 0 , t 0 ). Let κ > κ 0 and N be the integer defined by

M N κ 0 < κ ≤ M N +1 κ 0 ⇐⇒ N := 1 ln M ln κ κ0 -1.
Here ⌈r⌉ denotes, for r ∈ R, the smallest integer not smaller than r. Then by using iteratively (3.44) given by Step 3, we deduce that

|G - θ,H0 (x 0 , t 0 ) \ A κ | ≤ (1 -ση) N |G - θ,H0 (x 0 , t 0 ) \ A κ0 |.
By inserting the value of N , we conclude that

|G - θ,H0 (x 0 , t 0 ) \ A κ | ≤ 1 1 -ση |G - θ,H0 (x 0 , t 0 )| κ κ 0 -ε
, By combining (3.41) and (3.42) (see Step 2), we come back to the cube C R (0, T 0 ) and obtain the desired estimate.

Step 5 (Covering argument). We cover Q 1/2 by

Q 1/2 0, -1 4 ⊆ 1≤i≤N C R (X i , T i ) ⊆ Q 1 where the parabolic cylinders C R (X i , T i ) := C R (0, T 0 ) + (X i , T i ) -(0, T 0 ) are dis- joint. Then Q 1/2 0, -1 4 \ A κ ≤ N i=1 |C R (X i , T i ) \ A κ |.

By applying

Step 4 to each parabolic cylinder C R (X i , T i ), we get

Q 1/2 0, -1 4 \ A κ ≤ 1 ξ(1 -ση) N i=1 |C R (X i , T i )| κ κ 0 -ε . Since N i=1 |C R (X i , T i )| ≤ |Q 1 |
, we get the estimate given by (3.38).

Proof of Theorem 1.2

To prove Theorem 1.2, we differentiate the equation to obtain the result from the parabolic W 2,ε estimate obtained in Section 3. In the elliptic case, the proof of the elliptic W 3,ε estimate strongly uses the C 1,α estimates in order to apply W 2,ε estimates on the components of the gradient Du. In the parabolic case, the C 1,α estimates do not imply that u is differentiable with respect to the time variable. Thus the main new challenge which arises is to upgrade the regularity with respect to time, which is accomplished in Proposition 4.1.

The idea of the proof of Proposition 4.1 is to separate time and space in order to gain local regularity. First we are going to obtain regularity in space for a fixed time s, by applying the W 2,ε estimate on the derivatives of u. This step gives good quadratic approximations in space and, since u solves the PDE, we obtain a first-order approximation of the solution with respect to time. Instead of obtaining directly the estimate like in the elliptic case (see [START_REF] Armstrong | Partial regularity of solutions of fully nonlinear, uniformly elliptic equations[END_REF]), we proceed by contradiction by considering a local maximum (or minimum) and we use viscosity solution arguments and the uniform ellipticity of the operator.

Note that we use u is solution of the PDE to obtain a connection between Ψ(u) and Θ(u xi ), 1 ≤ i ≤ d, which is different than the elliptic case. Proposition 4.1. Assume that F satisfies (F1), F (0) = 0 and g(0, 0) = 0. Suppose that u ∈ C(Q 1 ) is a viscosity solution of (1.3) in Q 1 such that sup Q1 |u| ≤ 1. There exists a universal constant C 1 > 0 such that, for every κ

= (κ 1 , • • • , κ d ) ∈ R d + such that |κ| ≥ 1, d i=1 (y, s) ∈ Q 1/2 0, -1 4 : Θ(u xi , Q 1 )(y, s) ≤ κ i ⊆ (y, s) ∈ Q 1/2 0, -1 4 : Ψ(u, Q 1 )(y, s) ≤ C 1 1 + g C 0,1 (Q1) |κ| . Proof. Fix κ = (κ 1 , • • • , κ d ) ∈ R d + and (x 0 , t 0 ) ∈ Q 1/2 0, -1 4 such that, for every 1 ≤ i ≤ d, Θ(u xi , Q 1 )(x 0 , t 0 ) ≤ κ i . For all 1 ≤ i ≤ d, we select p i ∈ R d such that, for every y ∈ B 1 and -1 < s ≤ t 0 , |u xi (y, s) -u xi (x 0 , t 0 ) -p i • (y -x 0 )| ≤ κ i 1 2 |x 0 -y| 2 + t 0 -s . (4.1)
Define the polynomial approximation P to u given by

P (y, s) = u(x 0 , t 0 ) + b(s -t 0 ) + Du(x 0 , t 0 ) • (y -x 0 ) + 1 2 (y -x 0 ), M (y -x 0 ) (4.
2) where b and M are going to be chosen. First we set M := (p i j ) ∈ M d . Then to fix b, up to replacing M by M+M ⊤ 2 in (4.2), we can assume that M ∈ S d and then we take b := -F (M ) + g(x 0 , t 0 ). (4.3) To estimate the difference between P and u, we separate the difference into two parts, a space term and a time term

|u(y, s)-P (y, s)| ≤ |u(y, s)-u(x 0 , s)-Du(x 0 , t 0 )•(y-x 0 )-1 2 (y-x 0 ), M (y-x 0 ) | + |u(x 0 , s) -u(x 0 , t 0 ) -b(s -t 0 )|.
We proceed in eight steps. Steps 1-2 provide the upper bound on the space term. Then by introducing an adequate test function and arguing by a comparison principle argument, Steps 3-7 will give the corresponding upper bound on the time term. Finally, we will conclude in Step 8.

Step 1. We claim that for all 1 ≤ i ≤ d, (y, s) ∈ Q 1 with s ≤ t 0 ,

|u xi (y, s) -u xi (y, t 0 )| ≤ κ i |x 0 -y| 2 + t 0 -s . (4.4)
To prove this, notice that

|u xi (y, t 0 ) -u xi (x 0 , t 0 ) -p i • (y -x)| ≤ κi 2 |x 0 -y| 2 .
By combining this inequality with (4.1), we get

|u xi (y, s) -u xi (y, t 0 )| ≤ |u xi (y, s) -u xi (x 0 , t 0 ) -p i • (y -x 0 )| + |u xi (y, t 0 ) -u xi (x 0 , t 0 ) -p i • (y -x 0 )| ≤ κ i 1 2 |x 0 -y| 2 + t 0 -s + κi 2 |x 0 -y| 2 .
The proposed estimate (4.4) directly follows.

Step 2. We next prove the following "slice estimate": for every (y, s) ∈ Q 1 such that -1 < s ≤ t 0 , we have

u(y, s) -u(x 0 , s) -Du(x 0 , t 0 ) • (y -x 0 ) -1 2 (y -x 0 ), M (y -x 0 ) ≤ |κ| 7 6 |y -x 0 | 3 + √ 2 3 |s -t 0 | 3/2 . (4.5)
Since u ∈ C 1 with respect to the space variable, we can write

I : = u(y, s) -u(x 0 , s) -Du(x 0 , t 0 ) • (y -x 0 ) -1 2 (y -x 0 ), M (y -x 0 ) = (y -x 0 ) • ˆ1 0 Du(x 0 + τ (y -x 0 ), s) -Du(x 0 , t 0 ) -τ M (x 0 -y 0 )dτ .
It is clear that I ≤ I 1 + I 2 where I 1 and I 2 respectively denote

I 1 := (y -x 0 ) • ˆ1 0 Du(x 0 + τ (y -x 0 ), s) -Du(x 0 + τ (y -x 0 )), t 0 )dτ and
I 2 := (y -x 0 ) • ˆ1 0 Du(x 0 + τ (y -x 0 ), t 0 ) -Du(x 0 , t 0 ) -τ M (x 0 -y 0 )dτ .
It remains to determine some upper bounds on I 1 and I 2 . For I 1 , applying successively Cauchy-Schwarz inequality and Step 1 yields that

I 1 ≤ |y -x 0 | ˆ1 0 |Du(x 0 + τ (y -x 0 ), s) -Du(x 0 + τ (y -x 0 )), t 0 )| dτ ≤ |y -x 0 | ˆ1 0 |κ|(τ 2 |y -x 0 | 2 + t 0 -s)dτ ≤ |κ| 1 3 |y -x 0 | 3 + |y -x 0 ||s -t 0 | .
We estimate I 2 by using (4.1) and the same computations than those used in [START_REF] Armstrong | Partial regularity of solutions of fully nonlinear, uniformly elliptic equations[END_REF]. For sake of completeness and reader convenience, we give here the arguments. According to (4.1),

|u xi (x 0 + τ (y -x 0 ), t 0 ) -u xi (x 0 , t 0 ) -τ p i • (y -x 0 )| ≤ 1 2 κ i τ 2 |y -x 0 | 2 .
By Cauchy-Schwarz inequality,

I 2 ≤ |y -x 0 | ˆ1 0 1 2 d i=1 κ 2 i 1/2 τ 2 |y -x 0 | 2 dτ = |κ| 6 |y -x 0 | 3 .
To obtain (4.5), it suffices to apply Young inequality which yields

|y -x 0 ||s -t 0 | ≤ 2 3 |y -x 0 | 3 + √ 2 3 |s -t 0 | 3/2 ,
and a simple calculation gives (4.5).

To complete the proof of Proposition 4.1, it remains now to get a similar estimate for the term in time. We will show that there exists a universal constant

C 2 ≥ 1 such that, for all -1 < s ′ ≤ t 0 , |u(x 0 , s ′ ) -u(x 0 , t 0 ) -b(s ′ -t 0 )| ≤ C 2 1 + g C 0,1 (Q1) |κ||t 0 -s ′ | 3/2 . ( 4.6)
We are going to prove that, for all -1 < s ′ ≤ t 0 ,

u(x 0 , s ′ ) -u(x 0 , t 0 ) -b(s ′ -t 0 ) ≥ -C 2 1 + g C 0,1 (Q1) |κ||t 0 -s ′ | 3/2 , (4.7)
the argument for the reverse inequality is entirely parallel. To do this, we will consider a suitable test function taking into account the size of b.

Step 3. We claim that the following a priori bound on b = -F (M ) + g(x 0 , t 0 ) holds

|b| ≤ c 2 |κ| + 1 + g C 0,1 (Q1) , (4.8)
where c 2 > 0 a universal constant. By considering (4.1) for s = t 0 and selecting y ∈ B 3/4 ⊆ B 1 such that x 0 -y = 1 4

p i |p i | if p i = 0, we get 1 4 |p i | -sup y∈B 3/4 |u xi (y, t 0 ) -u xi (x 0 , t 0 )| ≤ |u xi (y, t 0 ) -u xi (x 0 , t 0 ) -p i • (x 0 -y)| ≤ κi 2 |x 0 -y| 2 .
By rearranging the terms, we obtain

1 4 |p i | ≤ 9 32 κ i + sup y∈B 3/4 |u xi (y, t 0 ) -u xi (x 0 , t 0 )| ≤ 9 32 κ i + 2 sup Q 3/4
|Du|.

Now by Proposition 2.1, there exists a universal constant C > 0 such that sup

Q 3/4 |Du| ≤ C 1 + g C 0,1 (Q1) .
We deduce that there exists a universal constant c > 9/8 such that, for all 1 ≤ i ≤ d,

|p i | ≤ c κ i + 1 + g C 0,1 (Q1) . If M denotes the (L 2 , L 2 )-norm of M , i.e. M = sup |x|=1 |M x|, then our choice for M = (p i j ) implies that M ≤ dc |κ| + 1 + g C 0,1 (Q1) .
Since F satisfies (F1) and F (0) = 0, it is immediate to check that for all N ∈ S d , |F (N )| ≤ dΛ N . Thus, by using (4.3), g(0, 0) = 0 and (x 0 , t 0 ) ∈ Q 1 , we get

|b| ≤ dΛ M + |g(x 0 , t 0 )| ≤ dΛ M + g C 0,1 (Q1) ,
and the claim (4.8) easily follows.

Step

4. Let s ′ ∈ (-1, t 0 ]. Next we show that φ ≥ u holds on ∂ p B 3/4 × {s ′ < s ≤ t 0 } ,
where we have defined

φ(y, s) := u(x 0 , s ′ ) + Du(x 0 , t 0 ) • (y -x 0 ) + 1 2 (y -x 0 ), M (y -x 0 ) + b + C0β 4 + 2 g C 0,1 (Q1) |κ||s ′ -t 0 | 1/2 (s-s ′ )+ C0 6 |κ|(|y -x 0 | 3 +|s ′ -t 0 | 3/2 )
where β := max{1, 2Λ(d + 1)} and C 0 is a constant depending on d, λ, Λ and g C 0,1 (Q1) given by

C 0 := 768 c 2 1 + g C 0,1 (Q1) + 3 . (4.9) A direct computation yields φ(x 1 , t 1 )-φ(x 0 , t 1 ) = Du(x 0 , t 0 )•(x 1 -x 0 )+ 1 2 (x 1 -x 0 ), M (x 1 -x 0 ) + C0 6 |κ||x 1 -x 0 | 3 . After rearranging the terms, we obtain u(x 1 , t 1 ) -u(x 0 , t 1 ) -Du(x 0 , t 0 ) • (x 1 -x 0 ) -1 2 (x 1 -x 0 ), M (x 1 -x 0 ) ≥ C0 6 |κ||x 1 -x 0 | 3 . (4.10) Moreover, we know by (4.5) that |u(x 1 , t 1 ) -u(x 0 , t 1 ) -Du(x 0 , t 0 ) • (x 1 -x 0 ) -1 2 (x 1 -x 0 ), M (x 1 -x 0 ) | ≤ |κ| 7 6 |x 1 -x 0 | 3 + √ 2 3 |t 1 -t 0 | 3/2
. By using (4.10) we deduce from the inequality above that

|κ| 7 6 |x 1 -x 0 | 3 + √ 2 3 |t 1 -t 0 | 3/2 ≥ C0 6 |κ||x 1 -x 0 | 3 .
Since C0 6 ≥ 7 3 , we obtain

|x 1 -x 0 | ≤ 2 √ 2 7 1/3 |t 1 -t 0 | 1/2 .
In particular, this yields the desired claim.

Step 6. We next show that u-φ cannot achieve any local maximum in the cylinder Q given by Q

:= {(y, s) ∈ B 1 × [s ′ , t 0 ] : |y -x 0 | < |s ′ -t 0 | 1/2 },
by arguing that φ is a strict supersolution in Q i.e.,

∂ t φ + F (D 2 φ) > g, in Q. ( 4.11)
We verify (4.11) by a direct computation. By the uniform ellipticity condition (F1), and noticing that the perturbation is a positive matrix, This confirms (4.11).

F M + C 0 |κ| 2 |y -x 0 | I d + y -x 0 |y -x 0 | ⊗ y -x 0 |y -x 0 | ≥ F (M )- Λ(d + 1) 2
Step 7. By Steps 4-6, uφ cannot achieve any positive global maximum. Hence,

u ≤ φ in B 3/4 × [s ′ , t 0 ].
In particular, for s = t 0 , u(x 0 , t 0 ) ≤ φ(x 0 , t 0 ), this yields u(x 0 , t 0 ) ≤ u(x 0 , s ′ )+b(t 0 -s ′ )+ C0β 4 + 2 g C 0,1 (Q1) |κ||t 0 -s ′ | 3/2 + C0 6 |κ||t 0 -s ′ | 3/2 . After rearranging the terms, we get

-C 0 1+β 4
+ 2 g C 0,1 (Q1) |κ||t 0s ′ | 3/2 ≤ u(x 0 , s ′ )u(x 0 , t 0 )b(s ′t 0 ). Thus, by using (4.9), it follows (4.7) by taking C 2 := 192(1 + β)(c 2 + 3).

Step 8. We conclude the argument. By combining both (4.5) and (4.6), we deduce that, for all (y, s) ∈ By setting C 1 := C 2 + 7 6 , this implies Ψ(u, Q 1 )(x 0 , t 0 ) ≤ C 1 1 + g C 0,1 (Q1) |κ|.

Now we can give the proof of the parabolic W 3,ε estimate stated in the introduction.

Proof of Theorem 1.2. If u ≡ 0 on Q 1 , then the estimate is clear, so we may assume that sup Q1 |u| > 0. In Step 1, we reduce the proof of the theorem to the case of sup Q1 |u| ≤ 1, g(0, 0) = 0 and F (0) = 0 by scaling arguments. In Step 2, we prove the theorem under these assumptions.

Step 1. We first reduce to the case that g(0, 0) = 0. If g(0, 0) = 0, define g(x, t) := g(x, t)g(0, 0) and u(x, t) := u(x, t)tg(0, 0) is a solution of By applying (4.14) and using (4.12), we obtain (x, t) ∈ Q 1/2 0, - Noticing that Ψ( u, Q 3/4 )(x, t) = 1 β Ψ(u, Q 3/4 )(x, t), we obtain that, for all κ > 0, (x, t) ∈ Q 1/2 0, -1 4 : Ψ(u, Q 3/4 )(x, t) > βκ ≤ C κ 1 + g C 0,1 (Q1) /β -ε . This is equivalent to (4.14).

∂ t u + F (D 2 u) = g.
Step 2. Assuming that sup Q1 |u| ≤ 1, g(0, 0) = 0 and F (0) = 0, we give the proof of (4.13). It suffices to get the inequality for κ ≥ κ 1 , where κ 1 is a universal constant. According to Proposition 2.1, Du is continuous and there exists a constant C such that sup

Q 3/4 |Du| ≤ C 1 + g C 0,1 (Q1) . (4.15)
Moreover, we claim that for every unit direction e ∈ R d , |e| = 1, the function u e = e • Du satisfies the inequalities ∂ t u e + P - λ,Λ (u e )g C 0,1 (Q1) ≤ 0 ≤ ∂ t u e + P + λ,Λ (u e ) + g C 0,1 (Q1) , in Q 1 , in the viscosity sense. We refer to [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF]Lemma 3.12] for the elliptic version of this statement which is easy to generalize to the parabolic setting. According to Proposition 3.1, we have, for each κ > 0, |{(x, t) ∈ Q 1/2 0, - This completes the proof of (4.13).

2. 1 .

 1 Hypotheses and notation. Let S d ⊆ M d be the set of symmetric matrices. If M ∈ M d , M ⊤ denotes the transpose of M . Recall that the Pucci extremal operators are defined for constants 0 < λ ≤ Λ and M ∈ S d by P + λ,Λ (M ) := sup λI d ≤A≤ΛI d -tr(AM ), and P - λ,Λ (M ) := inf λI d ≤A≤ΛI d -tr(AM ).

  holds true and we have |V | ≤ (3.10)

  1 + τ ) b+1 e aθ -1 . (3.14)We next show that, with this choice of parameters, φ is a nonnegative subsolution in G θ,1+τ (0, -τ ) which vanishes on the lateral boundary of G θ,1+τ (0, -τ ) and is not too large initially. This plays the role of the "bump function" from the elliptic case [4, Lemma 4.1]. Lemma 3.4. Let τ > 0. For a, b and c given by (3.12)-(3.14), the function φ satisfies

r 2 .

 2 Recalling (3.18), we obtain (3.23).

2 + 2 +

 22 C 0 |κ||y-x 0 |.This inequality yieldsb + 2 g C 0,1 (Q1) + β 4 C 0 |κ||s ′t 0 | 1/F M + C 0 |κ| 2 |yx 0 | I d + yx 0 |yx 0 | ⊗ yx 0 |yx 0 | ≥ b+2 g C 0,1 (Q1) |κ||s ′ -t 0 | 1/2 +F (M )+ β 4 C 0 |κ||s ′ -t 0 | 1/2 -Λ(d + 1) 2 C 0 |κ||y-x 0 |.For our choice given by (4.3), we have b + F (M ) = g(x 0 , t 0 ). Recalling the value of β and |κ| ≥ 1, we obtain that the following inequality holds inQ: b + 2 g C 0,1 (Q1) + β 4 C 0 |κ||s ′t 0 | 1/F M + C 0 |κ| 2 |yx 0 | I d + yx 0 |yx 0 | ⊗ yx 0 |yx 0 | > g C 0,1(Q1) |yx 0 | + |st 0 | 1/2 + g(x 0 , t 0 ) ≥ g(y, s).

Q 1 , 2 3 2 ≤

 122 s ≤ t 0 , |u(y, s)-P (y, s)| ≤ |κ| 7 6 |yx 0 | 3 + √ |st 0 | 3/2 + C 2 1 + g C 0,1 (Q1) |κ||t 0 -s| 3/C 2 1 + g C 0,1 (Q1) + 7 6 |κ| |yx 0 | 3 + |st 0 | 3/2 .

  0)| ≤ sup Q1 |u| + |g(0, 0)|.Then we reduce to the case that F (0) = 0. If F (0) = 0, then, by ellipticity, there exists a ∈ R such thatF (aI d ) = 0 with |a| ≤ 1 λd |F (0)|. (4.12)Define the operator F (M ) := F (M + aI d ) and observe that F satisfies (F1) with the same ellipticity constants λ, Λ andF (0) = F (aI d ) = 0. It is clear that u(x, t) := u(x, t) -1 2 a|x| 2 is a solution of ∂ t u + F (D 2 u)

1 4 :

 4 Ψ(u, Q 3/4 )(x, t) > κ ≤ C κ sup Q1 |u| + 1 2λd |F (0)| -ε, and we get the inequality given by Theorem 1.2.Next we reduce to the case that sup Q1 |u| ≤ 1. Assume that we have shown if sup Q1 |u| ≤ 1 and F (0) = 0, then for all κ > 0,(x, t) ∈ Q 1/2 0, -1 4 : Ψ(u, Q 3/4 )(x, t) > κ ≤ C κ 1 + g C 0,1 (Q1) -ε . (4.13)We claim that if F (0) = 0, and β := sup Q1 |u| > 0 then for all κ > 0,(x, t) ∈ Q 1/2 0, -1 4 : Ψ(u, Q 3/4 )(x, t) > κ ≤ C κ β + g C 0,1 (Q1) -ε . (4.14)Define the function g := (1/β)g, the operator F (M ) := β -1 F (βM ) and observe g ∈ C 0,1 (Q 1 ) with g C 0,1 (Q1) = g C 0,1 (Q1) /β, F satisfies (F1) with the same ellipticity constants λ, Λ andF (0) = β -1 F (0) = 0. It is clear that u := u/β is a solution of ∂ t u + F (D 2 u) = g with sup Q1 | u| = 1.By applying (4.13) to u, we obtain that, for all κ > 0, (x, t) ∈ Q 1/2 0, -1 4 : Ψ( u, Q 3/4 )(x, t) > κ ≤ C κ 1 + g C 0,1 (Q1) /β -ε .

1 4 :

 4 Θ(u e , Q 3/4 )(x, t) > κ}| ≤ C κ sup Q 3/4 |u e | + g C 0,1 (Q1) -ε, where C, ε > 0 are universal constants. Thus, we deduce from (4.15) that there exists a new universal constant C > 0 such that for all κ > 0,|{(x, t) ∈ Q 1/2 0, -1 4 : Θ(u e , Q 3/4 )(x, t) > κ}| ≤ C κ 1 + g C 0,1 (Q1) By Proposition 4.1, there exists a universal constant C 1 such that, for all κ > C 1 1 + g C 0,1 (Q1) , (x, t) ∈ Q 1/2 0, -1 4 : Ψ(u, Q 3/4 )(x, t) > κ t) ∈ Q 1/2 0, -1 4 : Θ(u xi , Q 3/4 )(x, t) > κ C 1 1 + g C 0,1(Q1) we obtain the desired result for κ > C 1 1 + g C 0,1 (Q1) by applying (4.16).

  .22) To obtain the first assertion, it suffices to show that the vertex(x 2 , t 2δh) of G 2 is in Q r/4 (x 2 , t 2 ) and G 2 ∩ {(y, s) : s = t 2 } ⊆ B r/4 (x 2 ).

							First, by using (3.19)
	and (3.20), the inequality
	δh ≤	1 16ν 2	α θ	θν 2 r 2 =	α 16	r 2 ≤	r 2 16

implies (x 2 , t 2δh) ∈ Q r/4 (x 2 , t 2 )

. By the definition of the parabolic ball G 2 and using

(3.19

) and (3.20), each (z, t 2

  t2-s2 (y 2 , s 2 ), τ ≤ t .By(3.27), the second infimum in the right-hand side above is nonnegative for t ∈ [s 2 , t 2 ] and it suffices to study the sign of ζ(t). First notice ζ(s 2 ) = χ(y 2 , s 2 ) ≥ 0 by (3.28). Then, since (z 2 , t 2 ) ∈ G 1/2,t2-s2 (y 2 , s 2 ) we obtain also by (3.28) thatζ(t 2 ) ≤ χ(z 2 , t 2 ) < 0.By (3.29), we deduce there exists a time t 3 ∈ [s 2 , t 2 ) such thatζ(t 3 ) = inf χ(z, τ ) : (z, τ ) ∈ G 1/2,t2-s2 (y 2 , s 2 ) and τ ≤ t 3 = 0.Since χ ∈ LSC(Q 1 ), there exists (z 3 , t 3 ) ∈ G 1/2,t2-s2 (y 2 , s 2 ) realizing this infimum, and so, satisfying (3.26). Then, the inclusion G 1/2,t2-s2 (y 2 , s 2 ) ⊆ Q r (x 2 , t 2 ) yields the claim. To complete the proof of (3.26), it remains to check (3.27), (3.28) and (3.29). To get (3.27), observe that, for all

First we claim that φ ≥ u on B 3/4 × {s ′ }. To prove this, it follows from (4.5) that for all y ∈ B 3/4 ⊆ B 1 , u(y, s ′ ) ≤ u(x 0 , s ′ ) + Du(x 0 , t 0 ) • (y

, this yields that

Then we claim that φ ≥ u on {(y, s) : |y| = 3/4, s ′ < s ≤ t 0 }. Arguing by contradiction, assume that there exists (x 1 , t 1 ) with

Neglecting the nonnegative time terms, we obtain

By combining the slice estimate (4.5) and the bound on b given by (4.8), we obtain that

the inequality above simply reduces to C0 384 |κ| < c 2 1 + g C 0,1 (Q1) + 3 (|κ| + 1). By inserting (4.9), we obtain 2|κ| < |κ| + 1, and we get a contradiction if |κ| ≥ 1.

Step 5. We claim that: u-φ can attain a positive global maximum only in (y, s) : |y -

By Step 4, we may assume that u-φ attains a positive global maximum at (x 1 , t 1 ) ∈ B 3/4 × {s ′ < s ≤ t 0 }: (uφ)(x 1 , t 1 ) = sup (uφ)(y, s).

In particular, (uφ)(x 1 , t 1 ) ≥ (uφ)(x 0 , t 1 ).