Remarks on nonlinear uniformly parabolic equations

Abstract

This paper provides a numb e r o f w orking tools for the discussion of fully nonlinear parabolic equations. These include: a proof that the maximum principle which provides L 1 estimates of strong" solutions of extremal equations by L n+1 norms of the forcing term over the contact set" remains valid for viscosity solutions in an L n+1 sense, a gradient estimate in L p for p n + 1 n + 2 for solutions of extremal equations with forcing terms in L n+1 , the use of this estimate in improving the range of p for which the maximum principle rst alluded to holds obtaining some p n + 1 -but without the contact set , a proof of the strong solvability of Dirichlet problems for extremal equations with forcing terms in L p for some p n + 1 , and the twice parabolic di erentiability a.e. of W 2;1;p functions for n + 2 = 2 p . 0. Introduction. In this work we provide a number of tools for the discussion of nonlinear parabolic equations under appropriate structure conditions. In particular, we ll gaps in the current literature and thereby prove a full generalization of the maximum principle" see below to viscosity solutions of certain extremal fully nonlinear equations with measurable forcing terms. While going about this task, we m ust resolve certain existence questions, and this is done as well. The results obtained are formulated in terms of standard extremal equations so that they apply to many other equations. In addition, some new proofs of known to varying degrees results are given.Let 0 be constants and de ne P , X = , trace X + + trace X , for X 2 S n , the set of real symmetric n n matrices. Here trace X + respectively, trace X , is the sum of the positive eigenvalues of X respectively, ,X .Let 0 be another constant. We discuss several results concerning solutions of the parabolic inequality 0:1 L u : = u t + P , D 2 u , jDuj f as well as the equation L u = f in Q = 0; T , where is an open, bounded domain in IR n , n 2. Here Ducorresponds to the spatial gradient u x 1 ; : : : ; u x n , D 2 u corresponds to the spatial Hessian matrix u xi;xj and jpj is the Euclidean 1 length of p 2 IR n . The notation P , " re ects the fact that this function was introduced by Pucci as well as the property that P , X is the minimum of ,trace AX o v er A 2 S n with I A I. The extremal parabolic equations L u = f w ere rst studied by Astesiano 1 ; for the modern theory of nonlinear parabolic equations we refer to Krylov 24 and Lieberman 26 .The rst version of the maximum principle of concern to us is stated below for easy reference. It follows immediately from Tso's work 32 which re ned a result of Krylov 23 see also the contemporaneous works of Reye 30 and Nazarov and Ural'tseva 29 , and a recent paper of Cabre 3 . In the statement and hereafter, the parabolic boundary of Q is denoted by @ p Q, the diameter of b y d , and the diameter of Q by d Q : where C = C 1= ; n; jQj n+1 =d depends only on the indicated quantities and is bounded for bounded a r guments.Hereafter we refer to the inequality 0.4 as the maximum principle". Ge...

Isolated Singularities for Fully Nonlinear Elliptic Equations

“…We say that We refer to [20] for the theory of viscosity solutions and to [6,37] for L p -viscosity solutions of the general equations F(x, u, Du, D 2 u)=0, with respectively continuous and measurable dependence on x. A complete treatment of the parabolic counterpart can be found in [12].…”

Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE

“…We refer the reader to [12], Corollary 2 for the precise statement of the maximum principle, to [1,8,14] for the derivation of p 0 , and to [4,9,11] and [21] for more on the theory of of L p -viscosity solutions. We refer the reader to [12], Corollary 2 for the precise statement of the maximum principle, to [1,8,14] for the derivation of p 0 , and to [4,9,11] and [21] for more on the theory of of L p -viscosity solutions.…”

Regularity Results for a Class of Semilinear Parabolic Degenerate Equations and Applications

“…In this section, we derive some consequences from the results of the previous sections. Let us recall first from [6] a useful result about Sobolev spaces W 2,1,p loc (R N ×(0, T )), i.e. : the space of the functions u, such that u, ∇u, ∇ 2 u, ∂ t u ∈ L p loc (R N × (0, T )).…”

“…For the proof we refer the reader to [6]. If the assumptions of Theorem 2.5 are satisfied, then we can apply Proposition 5.2 to u, obtaining that for a.e.…”