/pdf/asymptotic-analysis-of-periodic-structures-14wjonu4ba.pdf

Asymptotic Analysis of Periodic Structures

DGM: A deep learning algorithm for solving partial differential equations

The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academics in mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises for the reader.

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The Porous Medium Equation: Mathematical Theory

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

We provide existence, comparison and stability results for one- dimensional backward stochastic differential equations (BSDEs) when the coefficient (or generator) $F(t,Y, Z)$ is continuous and has a quadratic growth in $Z$ and the terminal condition is boundede also give, in this framework, the links between the solutions of BSDEs set on a diffusion and viscosity or Sobolev solutions of the corresponding semilinear partial differential equations

/pdf/backward-stochastic-differential-equations-and-partial-5f6lw1axlk.pdf

Backward stochastic differential equations and partial differential equations with quadratic growth

Non-linear elliptic and parabolic equations involving measure data

/pdf/an-l-1-theory-of-existence-and-uniqueness-of-solutions-of-2r7m2ktbta.pdf

An $L^1$ -theory of existence and uniqueness of solutions of nonlinear elliptic equations

Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations

We consider the differential problem (*) { A ( u ) = μ in Ω , u = 0 on ∂ Ω , where Ω is a bounded, open subset of R N , N ≥ 2, A is a monotone operator acting on W 0 1 , p ( Ω ) , p > 1, and μ is a Radon measure on Ω that does not charge the sets of zero p -capacity. We prove a decomposition theorem for these measures (more precisely, as the sum of a function in L 1 (Ω) and of a measure in W −1, p ′ (Ω)), and an existence and uniqueness result for the so-called entropy solutions of (*) .

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Lucio Boccardo

Papers

Non-linear elliptic and parabolic equations involving measure data

An $L^1$ -theory of existence and uniqueness of solutions of nonlinear elliptic equations

Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations

Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data

Nonlinear Parabolic Equations with Measure Data