Related papers: Optimal H\"{o}lder regularity for solutions to Sig…
This paper is devoted to a proof of optimal regularity, near the initial state, for weak solutions to the two-phase parabolic obstacle problem. The approach used here is general enough to allow us to consider the initial data belonging to…
We study certain obstacle type problems involving standard and nonlocal minimal surfaces. We obtain optimal regularity of the solution and a characterization of the free boundary.
In this work we consider an inhomogeneous two-phase obstacle-type problem driven by the fractional Laplacian. In particular, making use of the Caffarelli-Silvestre extension, Almgren and Monneau type monotonicity formulas and blow-up…
We establish optimal, quantitative H\"oder estimates for the gradient of solutions to a class of degenerate elliptic equations with Hamiltonian terms. The presence of such lower-order terms introduces additional challenges, particularly in…
This paper considers the time-harmonic Maxwell equations with impedance boundary condition.We present $H^2$-norm bound and other high-order norm bounds for strong solutions. The $H^2$-estimate have been derived in [M. Dauge, M. Costabel and…
We study the inverse problem of determining a Signorini obstacle from boundary measurements for the isotropic elasticity system. We prove that the obstacle can be uniquely determined by a single measurement of displacement and normal stress…
In this paper, we obtain local H\"older regularity for bounded, weak solutions to the anisotropic $p$-Laplace equation whose prototype structure is given by $$ \sum_{i=1}^N (|u_{x_i}|^{p_i-2}u_{x_i})_{x_i}=0,$$ where $1 < p_1 \leq p_2 \leq…
In this article, we communicate with the glimpse of the proofs of global regularity results for weak solutions to a class of problems involving fractional $(p,q)$-Laplacian, denoted by $(-\Delta)^{s_1}_{p}+(-\Delta)^{s_2}_{q}$, for $s_2,…
We investigate nonregular elliptic problems with boundary conditions of higher orders. We prove that these problems are Fredholm on appropriate pairs of inner product H\"ormander spaces that form a two-sided refined Sobolev scale. We also…
We consider a family of Leray-$\alpha$ models with periodic boundary conditions in three space dimensions. Such models are a regularization, with respect to a parameter $\theta$, of the Navier-Stokes equations. In particular, they share…
We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the…
We study the optimal lower and upper complexity bounds for finding approximate solutions to the composite problem $\min_x\ f(x)+h(Ax-b)$, where $f$ is smooth and $h$ is convex. Given access to the proximal operator of $h$, for strongly…
Consider a class of non-homogenous ultraparabolic differential equations with drift terms or lower order terms arising from some physical models, and we prove that weak solutions are H\"{o}lder continuous, which also generalizes the classic…
In this paper, we study the regularity assumptions commonly adopted in bilevel optimization with constrained lower-level problems, including the linear independence constraint qualification, the strict complementary slackness condition, and…
The objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic PDEs with measurable coefficients and the Liouville property at infinity. Initially, we address the…
We focus on constrained, $L$-smooth, potentially stochastic and nonconvex-nonconcave min-max problems either satisfying $\rho$-cohypomonotonicity or admitting a solution to the $\rho$-weakly Minty Variational Inequality (MVI), where larger…
We consider optimization problems with manifold-valued constraints. These generalize classical equality and inequality constraints to a setting in which both the domain and the codomain of the constraint mapping are smooth manifolds. We…
Based on a compactness method, we establish regularity criteria for suitable weak solutions to the surface growth model with a forcing term. These criteria imply that the H\"older regularity of solutions follows from smallness conditions on…
We study the regularity of solutions of elliptic second order boundary value problems on a bounded domain $\Omega$ in $\mathbb R^3$. The coefficients are not necessarily continuous and the boundary conditions may be mixed, i.e. Dirichlet on…
We study a class of semilinear free boundary problems in which admissible functions $u$ have a topological constraint, or spanning condition, on their 1-level set. This constraint forces $\{u=1\}$, which is the free boundary, to behave like…