Related papers: On the Time Derivative in an Obstacle Problem
The article addresses the convergence of implicit and semi-implicit, fully discrete approximations of a class of nonlinear parabolic evolution problems. Such schemes are popular in the numerical solution of evolutions defined with the…
In this paper, we consider the obstacle problem for the fractional Laplace operator $(-\Delta)^s$ in the Euclidian space $\mathbb{R}^n$ in the case where $1<s<2$. As first observed in \cite{Y}, the problem can be extended to the upper…
We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p$-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their…
For the N>=2 dimensional incompressible Naver-Stokes Equation, We have got its solution as a power series of time t, in which the coefficients are all known functions determined only by the initial velocity v0. We also prove that the…
We study the regularity of solutions to the obstacle problem for the parabolic biharmonic equation. We analyze the problem via an implicit time discretization, and we prove some regularity properties of the solution.
We consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in [8]. More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of…
In this paper a Pohozaev type inequality is stated for variable exponent Sobolev spaces in order to prove non existence of nontrivial weak solutions for a Dirichlet problem with non-standard growth. The obtained results generalize a…
We give an elementary proof of long time inviscid damping for Sobolev perturbations near the Couette flow $(y,0)$ for the 2D Euler equations on $\mathbb{T} \times \mathbb{R}$. For any $s>1$ and any initial vorticity perturbation of size…
Differential equations parameterized by neural networks become expensive to solve numerically as training progresses. We propose a remedy that encourages learned dynamics to be easier to solve. Specifically, we introduce a differentiable…
We consider a one-Laplace equation perturbed by $p$-Laplacian with $1<p<\infty$. We prove that a weak solution is continuously differentiable ($C^{1}$) if it is convex. Note that similar result fails to hold for the unperturbed one-Laplace…
In this paper we prove a theorem of global time-extension for the local classical solution of Navier-Stokes's evolution problem in $\Real^n$ with $n\geqslant2$ for incompressible fluids subjected to external forces and regular initial…
We establish the solvability of second order divergence type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be only measurable in one spatial direction on each small parabolic cylinder with the spatial…
We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed…
We report on a time regularity result for stochastic evolutionary PDEs with monotone coefficients. If the diffusion coefficient is bounded in time without additional space regularity we obtain a fractional Sobolev type time regularity of…
We prove a necessary optimality condition for isoperimetric problems on time scales in the space of delta-differentiable functions with rd-continuous derivatives. The results are then applied to Sturm-Liouville eigenvalue problems on time…
The existence and uniqueness of weak solutions to dynamical low-rank evolution problems for parabolic partial differential equations in two spatial dimensions is shown, covering also non-diagonal diffusion in the elliptic part. The proof is…
We study the parabolic $p$-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space-time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskii spaces and…
In this paper we consider a class of obstacle problems of the type %\begin{equation*} %\int_{\Omega}\left<A(x, Du), D(\varphi-u)\right> \, \dx\ge0\qquad\forall %\varphi\in W^{1,q}(\Omega) \quad {\mathrm{s.t.}} \quad \varphi \ge \psi…
The aim of this paper is to obtain the existence of solutions for the following fractional p-Laplacian Dirichlet problem with mixed derivatives \begin{eqnarray*}…
In this paper we study the limit as $p\to \infty$ in the evolution problem driven by the $p-$Laplacian with dynamical boundary conditions. We prove that the natural energy functional associated with this problem converges to a limit in the…