Related papers: On the free boundary for thin obstacle problems wi…
We study the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our…
We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity…
This paper investigates the regularity of solutions and structural properties of the free boundary for a class of fourth-order elliptic problems with Neumann-type boundary conditions. The singular and degenerate elliptic operators studied…
For bounded domains $\Omega$ with Lipschitz boundary $\Gamma$, we investigate boundary value problems for elliptic operators with variable coefficients of fourth order subject to Wentzell (or dynamic) boundary conditions. Using form…
We consider an elliptic-parabolic free boundary problem that models the fluid flow through a partially saturated porous medium. The free boundary arises as the interface separating the saturated and unsaturated regions. Our main goal is to…
We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class $A_p$…
We investigate general semilinear (obstacle-like) problems of the form $\Delta u = f(u)$, where $f(u)$ has a singularity/jump at $\{u=0\}$ giving rise to a free boundary. Unlike many works on such equations where $f$ is approximately…
We are concerned with the study of the existence and multiplicity of solutions for Dirichlet boundary value problems, involving the $( p( m ), \, q( m ) )-$ equation and the nonlinearity is superlinear but does not fulfil the…
In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set $\Omega \subset \mathbb R^n$, $n\geq 3$. Our proof relies on the discovery of effective monotonicity formulas holding along the…
The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds for stream functions as well as free-surface profiles and the total head are obtained under the…
We consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb R^{n+1}_+$ plus the area of the positivity set of that function in $\mathbb R^n$. We establish full…
We investigate the stability of timelike Ricci curvature lower bounds under low-regularity limits of Lorentzian metrics. Specifically, we prove that the synthetic curvature-dimension condition $TCD^e_p(K,N)$, which provides an optimal…
We give a short proof of the existence of a small piece of null infinity for $(3+1)$-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the…
In this paper, we prove a new continuous embedding theorem for fractional Sobolev spaces with variable exponents into variable exponent Lebesgue spaces on unbounded domains. As an application, we study a class of nonlocal elliptic problems…
For a compact spinc manifold $X$ with boundary $b_1(\partial X)=0$, we consider moduli spaces of solutions to the Seiberg-Witten equations in a generalized double Coulomb slice in $L^2_1$ (i.e., $W^{1,2}$) Sobolev regularity. We prove they…
This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schr\"odinger) equation $(\Delta + k^2 q) u = 0$ in a bounded domain for fixed non-resonance frequency $k>0$ and real-valued…
We prove the Lewy-Stampacchia inequalities for the two obstacles problem in abstract form for T-monotone operators. As a consequence for a general class of quasi-linear elliptic operators of Ladyzhenskaya-Uraltseva type, including…
The paper is devoted to the Hamiltonian treatment of classical and quantum properties of Liouville field theory on a timelike strip in 2d Minkowski space. We give a complete description of classical solutions regular in the interior of the…
We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of $$|\nabla u|^{\alpha}F(D^{2}u)=f $$ and, more generally, for viscosity supersolutions of $|\nabla u|^{\alpha}\,{M}^-_{\lambda,\Lambda}(D^{2}u)\le f$. The…
We study the boundary control of solutions of the Helmholtz and Maxwell equations to enforce local non-zero constraints. These constraints may represent the local absence of nodal or critical points, or that certain functionals depending on…