Related papers: On a Multiphase Vectorial Bernoulli Free Boundary …

We prove the local Lipschitz regularity of the minimizers of functionals of the form \[ \mathcal I(u)=\int_\Omega f(\nabla u(x))+g(x)u(x)\,dx\qquad u\in\phi+W^{1,1}_0(\Omega) \] where $g$ is bounded and $\phi$ satisfies the Lower Bounded…

We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers…

We study a Bernoulli type free boundary problem with two phases $$J[u]=\int_{\Omega}|\nabla u(x)|^2\,dx+\Phi\big({\mathcal M}_-(u), {\mathcal{M}}_+(u)\big), \quad u-\bar u\in W^{1,2}_0(\Omega), $$ where $\bar u\in W^{1,2}(\Omega)$ is a…

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 study the structure of solutions of the interior Bernoulli free boundary problem for $(-\Delta)^{\alpha/2}$ on an interval $D$ with parameter $\lambda > 0$. In particular, we show that there exists a constant $\lambda_{\alpha,D} > 0$…

In this paper we classify the nonnegative global minimizers of the functional \[ J_F(u)=\int_\Omega F(|\nabla u|^2)+\lambda^2\chi_{\{u>0\}}, \] where $F$ satisfies some structural conditions and $\chi_D$ is the characteristic function of a…

We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their $C^{1,\beta}$ regularity on the either side of the thin manifold, the optimal growth away from the free…

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 prove that nonnegative almost minimizers of the horizontal Bernoulli-type functional $$ J(u,\Omega):=\int_{\Omega}\Big(|\nabla_{\mathbb{G}} u(x)|^2+\chi_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz continuous in the intrinsic sense.

This paper is dedicated to the spectral optimization problem $$ \mathrm{min}\left\{\lambda_1^s(\Omega)+\cdots+\lambda_m^s(\Omega) + \Lambda \mathcal{L}_n(\Omega)\colon \Omega\subset D \mbox{ s-quasi-open}\right\} $$ where $\Lambda>0,…

In the present paper we establish the solvability of the Regularity boundary value problem in domains with (flat and Lipschitz) lower dimensional boundaries for operators whose coefficients exhibit small oscillations analogous to the…

We provide a rather complete description of the sharp regularity theory for a family of heterogeneous, two-phase variational free boundary problems, $\mathcal{J}_\gamma \to $ min, ruled by nonlinear, $p$-degenerate elliptic operators.…

We consider the optimization problem of minimizing $\int_{\Omega}G(|\nabla u|)+\lambda \chi_{\{u>0\}} dx$ in the class of functions $W^{1,G}(\Omega)$ with $u-\phi_0\in W_0^{1,G}(\Omega)$, for a given $\phi_0\geq 0$ and bounded.…

We consider triplets of densities $(u_1,u_2,u_3)$ minimizing the Dirichlet energy \[\sum_{j=1}^3 \int_{\Omega} |\nabla u_j|^2\,dx \] over a bounded domain $\Omega\subset \mathbb{R}^N$, subject to the partial segregation condition: \[…

We investigate partial regularity for vector valued local minimizers of double phase functionals, under vectorial obstacle type constraints satisfying appropriate topological properties.

This paper is devoted to a complete characterization of the free boundary of all solutions to the following spectral $k$-partition problem with measure and inclusion constraints: \[ \inf \left\{\sum_{i=1}^k \lambda_1(\omega_i)\; : \;…

We revisit the question of existence and regularity of minimizers to weighted least gradient problems on a fixed bounded domain, subject to a Dirichlet boundary condition, in the case where the boundary data is continuous and the weight…

We study the interior Bernoulli free boundary problem for the infinity Laplacian. Our results cover existence, uniqueness, and characterization of solutions (above a threshold representing the "infinity Bernoulli constant"), their…

We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general…

We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show…