Related papers: A Two-Phase Free Boundary Problem for Harmonic Mea…
We explore regularity properties of solutions to a two-phase elliptic free boundary problem near a Neumann fixed boundary in two dimensions. Consider a function u, which is harmonic where it is not zero and satisfies a gradient jump…
This paper is devoted to study the existence of solutions and the monotone method of second-order periodic boundary value problems when the lower and upper solutions $\alpha$ and $\beta$ violate the boundary conditions $…
The analysis and homogenization of a moving boundary problem for a highly heterogeneous, periodic two-phase medium is considered. In this context, the normal velocity governing the motion of the interface separating the two competing phases…
We propose a method to determine the smoothness of sufficiently flat solutions of one phase Hele-Shaw problems. The novelty is the observation that under a flatness assumption the free boundary --represented by the hodograph transform of…
We prove partial regularity of weakly stationary harmonic maps with (partially) free boundary data on manifolds where the domain metric may degenerate or become singular along the free boundary at the rate $d^\alpha$ for the distance…
We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries. This…
For the two-phase membrane problem $ \Delta u = {\lambda_+\over 2} \chi_{\{u>0\}} - {\lambda_-\over 2} \chi_{\{u<0\}} ,$ where $\lambda_+> 0$ and $\lambda_->0 ,$ we prove in two dimensions that the free boundary is in a neighborhood of each…
We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in super-critical dimensions. As a consequence of such a…
Since the seminal work of Kenig and Pipher, the Dahlberg-Kenig-Pipher (DKP) condition on oscillations of the coefficient matrix became a standard threshold in the study of absolute continuity of the harmonic measure with respect to the…
We prove boundedness, H\"older continuity, Harnack inequality results for local quasiminima to elliptic double phase problems of $p$-Laplace type in the general context of metric measure spaces. The proofs follow a variational approach and…
We prove Lipschitz continuity of viscosity solutions to a class of two-phase free boundary problems governed by fully nonlinear operators.
We are concerned with the problem of determining the nonlinear term in a semilinear elliptic equation by boundary measurements. Precisely, we improve [5, Theorem 1.3], where a logarithmic type stability estimate was proved. We show actually…
We consider a generalization of the Bernoulli free boundary problem where the underlying differential operator is a nonlocal, non-translation-invariant elliptic operator of order $2s\in (0,2)$. Because of the lack of translation invariance,…
In the recent work [DFM1, DFM2] G. David, J. Feneuil, and the first author have launched a program devoted to an analogue of harmonic measure for lower-dimensional sets. A relevant class of partial differential equations, analogous to the…
We establish sharp regularity for the value function, the pressure, and the free boundary in one-dimensional first-order mean field games with power coupling and compactly supported density. Under a standard nondegeneracy assumption on the…
We study boundary regularity of viscosity solutions to fully nonlinear degenerate or singular parabolic equations. The gradient-dependent degeneracy or singularity, along with the time derivative, introduces significant challenges beyond…
We study the regularity of the free boundary in one-phase Stefan problem with nonlinear operator. Using the Hodograph transform and a linearization technique, we prove that flat free boundaries are $C^{1, \alpha}$ in space and time. When…
We study solutions to a variational equation that models heat control on the boundary. This problem can be thought of as the two phase parabolic Signorini problem. We show that when the solution has a sign on the boundary, the study of the…
A new analytical formulation is prescribed to solve the Helmholtz equation in 2D with arbitrary boundary. A suitable diffeomorphism is used to annul the asymmetries in the boundary by mapping it into an equivalent circle. This results in a…
On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this…