Related papers: Gradient estimates for oblique derivative problems…
We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property. The primary advance compared to results…
In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods…
Boundary value problems for a class of quasilinear elliptic equations, with an Orlicz type growth and L^1 right-hand side are considered. Both Dirichlet and Neumann problems are contemplated. Existence and uniqueness of generalized…
We prove a version of the stochastic maximum principle, in the sense of Pontryagin, for the finite horizon optimal control of a stochastic partial differential equation driven by an infinite dimensional additive noise. In particular we…
Inspired by the penalization of the domain approach of Lions & Sznitman, we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal, possibly degenerate elliptic equations. Two different cases are considered:…
In this paper, we study quasilinear parabolic equations with the nonlinearity structure modeled after the $p(x,t)$-Laplacian on nonsmooth domains. The main goal is to obtain end point Calder\'on-Zygmund type estimates in the variable…
In this article, we introduce a new methodology to prove global parabolic Harnack inequalities on Riemannian manifolds. We focus on presenting a new proof of the global pointwise Harnack inequality satisfied by positive solutions of the…
In recent years a large literature on deep learning based methods for the numerical solution partial differential equations has emerged; results for integro-differential equations on the other hand are scarce. In this paper we study deep…
We study the oblique derivative problem for uniformly elliptic equations on cone domains. Under the assumption of axi-symmetry of the solution, we find sufficient conditions on the angle of the oblique vector for H\"older regularity of the…
In this paper we consider nonlinear parabolic systems with elliptic part which can be also degenerate. We prove optimal error estimates for smooth enough solutions. The main novelty, with respect to previous results, is that we obtain the…
We consider second-order elliptic equations in non-divergence form with oblique derivative boundary conditions. We show that any strong solutions to such problems are twice continuously differentiable up to the boundary provided that the…
We introduce a novel mesh-free and direct method for computing the shape derivative in PDE-constrained shape optimization problems. Our approach is based on a probabilistic representation of the shape derivative and is applicable for…
Variational methods based on optimization strategies are proposed to numerically solve a large family of nonlinear partial differential equations. They are all particular instances of gradient flows with general costs, including the…
Pointwise estimates for the gradient of solutions to the $p$-Laplace system with right-hand side in divergence form are established. They enable us to develop a nonlinear counterpart of the classical Calder\'on-Zygmund theory in terms of…
We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivation of this paper comes from the study on an abstract result in the…
A refined version of the strong maximum principle is proven for a class of second order ordinary differential equations with possibly discontinuous non-monotone nonlinearities. Then, exploiting this tool, some optimal regularity results…
In this paper, we study the statistical limits in terms of Sobolev norms of gradient descent for solving inverse problem from randomly sampled noisy observations using a general class of objective functions. Our class of objective functions…
In this paper we obtain regularity results for elliptic integro-differential equations driven by the stronger effect of coercive gradient terms. This feature allows us to construct suitable strict supersolutions from which we conclude…
Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) have a wide range of applications. In particular, high-dimensional PDEs with gradient-dependent nonlinearities appear often in the…
We provide a proof of strong maximum and minimum principles for fully nonlinear uniformly parabolic equations of second order. The approach is of parabolic nature, slightly differs from the earlier one proposed by L. Nirenberg and does not…