Related papers: Branch Points for (Almost-)Minimizers of Two-Phase…
In this paper, we study the regularity of the "regular" part of the free boundary for almost minimizers in the parabolic Signorini problem with zero thin obstacle. This work is a continuation of our earlier research on the regularity of…
We prove that the branching set of a solution to a two-dimensional two-phase Bernoulli problem with constant coefficients is locally finite. We do this via a Weierstrass representation formula, which allows to transform the problem into a…
We give a sufficient condition for H\"older continuity at a boundary point for quasiminima of double-phase functionals of $p,q$-Laplace type, in the setting of metric measure spaces equipped with a doubling measure and supporting a…
We establish interior regularity and optimal growth estimates for sign-changing minimizers of the $p-$singular or $p-$degenerate quasilinear Alt--Phillips functional throughout the full range of $1<p<\infty$ and of the nonlinearity power…
This doctoral thesis is devoted to the analysis of some minimization problems that involve nonlocal functionals. We are mainly concerned with the $s$-fractional perimeter and its minimizers, the $s$-minimal sets. We investigate the behavior…
Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational…
This paper is concerned with the two--phase obstacle problem, a type of a variational free boundary problem. We recall the basic estimates of Repin and Valdman (2015) and verify them numerically on two examples in two space dimensions. A…
In this paper we study vector-valued almost minimizers of the energy functional $$ \int_D\left(|\nabla\mathbf{u}|^2+2|\mathbf{u}|\right)\,dx . $$ We establish the regularity for both minimizers and the "regular" part of the free boundary.…
Cavity flow problems in two dimensions, as well as in the axially symmetric three-dimensional case, have been extensively studied in the literature from a qualitative perspective. While numerous results exist concerning minimizers or stable…
We approximate functionals depending on the gradient of $u$ and on the behaviour of $u$ near the discontinuity points, by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise…
This paper studies local properties of a two phase free boundary problem for the fractional Laplacian. The main result states that the two free boundaries of the positive and negativity sets cannot touch.
It is well-known that convex variational problems with linear growth and Dirichlet boundary conditions might not have minimizers if the boundary condition is not suitably relaxed. We show that for a wide range of integrands, including the…
We study vector-valued almost minimizers of the energy functional $$\int_D\left(|\nabla\mathbf{u}|^2+\frac2{1+q}\left(\lambda_+(x)|\mathbf{u}^+|^{q+1}+\lambda_-(x)|\mathbf{u}^-|^{q+1}\right)\right)dx,\quad0<q<1.$$ For H\"older continuous…
We study existence and structure of solutions to the Dirichlet and Neumann boundary problems associated with minimizers of the functional $I(u)=\int_{\Omega} (\phi(x, D u + F)+Hu) \, dx$, where $\phi (x, \xi)$, among other properties, is…
We find an explicit form of weak solutions to a Riemann problem for a degenerate semilinear parabolic equation with piecewise constant diffusion coefficient. It is demonstrated that the phase transition lines (free boundaries) correspond to…
We construct a branched center manifold in a neighborhood of a singular point of a $2$-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the…
In this paper we initiate the investigation of free boundary minimization problems ruled by general singular operators with $A_2$ weights. We show existence and boundedness of minimizers. The key novelty is a sharp $C^{1+\gamma}$ regularity…
We investigate existence and regularity properties of one-phase free boundary graphs, in connection with the question of whether there exists a complete non-planar free boundary graph in high dimensions.
For a fixed constant $\lambda > 0$ and a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ with $n \geq 2$, we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type…
We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers,…