DGM: A deep learning algorithm for solving partial differential equations

Multiple integrals in the calculus of variations and nonlinear elliptic systems

In this work we consider the problems
$$
\left\{\begin{array}{rcll}
\mathcal{L \,} u&=&f &\hbox{ in } \Omega,\\
u&=&0 &\hbox{ in } \mathbb{R}^N\setminus\Omega,
\end{array}
\right.
$$
and
$$
\left\{\begin{array}{rcll}
u_t +\mathcal{L \,} u&=&f &\hbox{ in } Q_{T}\equiv\Omega\times (0, T),\\
u (x,t) &=&0 &\hbox{ in } \big(\mathbb{R}^N\setminus\Omega\big) \times (0, T),\\
u(x,0)&=&0 &\hbox{ in } \Omega,
\end{array}
\right.
$$
where $\mathcal{L \,}$ is a nonlocal differential operator and $\Omega$ is a bounded domain in $\mathbb{R}^N$, with Lipschitz boundary.
 &nbsp
The main goal of this work is to study existence, uniqueness and summability of the solution $u$ with respect to the summability of the datum $f$.
In the process we establish an $L^p$-theory, for $p \geq 1$, associated to these problems and we prove some useful inequalities for the applications.

Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations

Evolution PDEs with Nonstandard Growth Conditions

with Ω a bounded open subset of R , N ≥ 2, p > 1, θ ≥ 0, and f a measurable function on whose summability we will make different assumptions. It is clear from the above example that the differential operator is defined on W 0 (Ω), but that it may not be coercive on the same space as u becomes large. Due to this lack of coercivity, standard existence theorems for solutions of nonlinear elliptic equations cannot be applied. In this paper, we will prove several existence and regularity results (depending on the summability of the datum f ) for the solutions of (1.1). Let us give the precise assumptions on the problems that we will study. Let Ω be a bounded open subset of RN , N ≥ 2. Let 1 < p < N, and let a : Ω × R × RN → RN be a Carathéodory function (that is, a(·, t, ξ) is measurable on Ω for every (t, ξ) in R × RN , and a(x, ·, ·) is continuous on R×RN for almost every x in Ω), such that the following assumptions hold: a(x, t, ξ) · ξ ≥ b(|t|) |ξ|p , (1.2)

/pdf/existence-results-for-nonlinear-elliptic-equations-with-4rdnj0y0u0.pdf

Existence results for nonlinear elliptic equations with degenerate coercivity

In this paper, we prove higher integrability results for the gradient of the solutions of some elliptic equations with degenerate coercivity whose prototype is$ - div{\left( {a{\left( {x,u} \right)}Du} \right)} = f\;in\;{D}\ifmmode{'}\else$'$\fi{\left( \Omega \right)},\;f \in L^{r} {\left( \Omega \right)},\;r > 1, $ where for example, a(x,u)=(1+|u|)−θ with θ ∈ (0,1). We study the same problem for minima of functionals closely related to the previous equation.

Elliptic Equations with Degenerate Coercivity: Gradient Regularity

Existence Results for Some Nonuniformly Elliptic Equations with Irregular Data

We prove existence, uniqueness, regularity results and estimates describing the behavior (both for large and small times) of a solution u of some nonlinear parabolic equations of Leray-Lions type including the p -Laplacian. In particular we show how the summability of the initial datum u 0 and the value of p influence the behavior of the solution u , producing ultracontractive or supercontractive estimates or extinction in finite time or different kinds of decay estimates.

Existence, uniqueness and behavior of solutions for a class of nonlinear parabolic problems

In this paper existence and regularity results for a class of degenerate nonlinear parabolic equations are proved. Indeed, the diffusion operator may degenerate as the solution diverges and may depend on space and time variables in a non–regular way, too. Some estimates on the behaviour of the solutions for t → +∞ are also given.

Parabolic equations with non-linear, degenerate and space-time dependent operators

Local regularity results for minima of functionals of the calculus of variation

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