In this short note, we study a system of PDEs with free boundaries inside the unit ball. In particular, we prove that solutions to our problem exist and, furthermore, that any solution must be symmetric.The core difficulty arises from the non-variational nature of the system, coupled with a highly singular right-hand side term. These characteristics preclude the application of standard techniques. However, the inherent symmetry of the problem, derived from the geometry, allows us to circumvent these challenges. Although we treat the Laplacian case in more detail, the approach is easily generalized to more general operators, m-component case, and p-Laplacian operator as stated in the paper.

Abstract: Given is a bounded domain C Rn, and a vector-valued function defined on ∂(depicting temperature distributions from different sources), our objective is to study the mathematical model of a physical problem of enclosing ∂with a specific volume of insulating material to reduce heat loss in a stationary scenario. Mathematically, this task involves identifying a vector-valued function u = (u1, um) (m ? 1) that represents the temperature within and gives rise to a free boundary, somehow reminiscent of, but not equivalent to, th Bernoulli free boundary problem.

We study a minimization problem with free boundary, resulting in hybrid quadrature domains for the Helmholtz equation, as well as some applications to inverse scattering problems.

In this article, we consider a weakly coupled p-Laplacian system of a Bernoulli-type free boundary problem, through minimization of a corresponding functional. We prove various properties of any local minimizer and the corresponding free boundary.

This paper concerns almost minimizers of the functional J(v,Ω)=∫Ω|Dv+|p+|Dv−|qdx,where 1<p≠q<∞ and Ω is a bounded domain of Rn, n≥1. We prove the universal Hölder regularity of local (1+ɛ)-minimizers, when ɛ is universally small. Moreover, we prove almost Lipschitz regularity of the local (1+ɛ)-minimizers, when |p−q|≪1 and ɛ≪1.

This paper revives a four-decade-old problem concerning regularity theory for (continuous) constraint maps with free boundaries. Dividing the map into two parts, the distance part and the projected image to the constraint, one can prove various properties for each component. As has already been pointed out in the literature, the distance part falls under the classical obstacle problem, which is well-studied by classical methods. A perplexing issue, untouched in the literature, concerns the properties of the projected image and its higher regularity, which we show to be at most of class C2,1. In arbitrary dimensions, we prove that the image map is globally of class W3,BMO, and locally of class C2,1 around the regular part of the free boundary. The issue becomes more delicate around singular points, and we resolve it in two dimensions. In the appendix, we extend some of our results to what we call leaky maps.

For a given constant lambda>0 and a bounded Lipschitz domain D subset of R-n(n >= 2), we establish that almost-minimizers of the functional J(v;D)=(D)integral(m)& sum;(i=1)|del vi(x)|(p)+lambda chi{|v|>0}(x) dx, 1<p<infinity, where v=(v(1),<middle dot><middle dot><middle dot>,v(m)),and m is an element of N , and , exhibit optimal Lipschitz continuity in compact sets of D. Furthermore, assuming p >= 2 and employing a distinctly different methodology, we tackle the issue of boundary Lipschitz regularity for <bold>v</bold>. This approach simultaneously yields an alternative proof for the optimal local Lipschitz regularity for the interior case.

The primary objective of this paper is to explore the multi-phase variant of quadrature domains associated with the Helmholtz equation, commonly referred to as k-quadrature domains. Our investigation employs both the minimization problem approach, which delves into the segregation ground state of an energy functional, and the partial balayage procedure, drawing inspiration from the recent work by Gardiner and Sj & ouml;din. Furthermore, we present practical applications of these concepts in the realms of acoustic waves and magnetic fields.

This paper investigates the possible scattering and nonscattering behavior of an anisotropic and inhomogeneous Lipschitz medium at a fixed wave number and with a single incident field. We connect the anisotropic nonscattering problem to a Bernoulli type free boundary problem. By invoking methods from the theory of free boundaries, we show that an anisotropic medium with Lipschitz but not C1,α boundary scatters every incident wave that satisfies a nondegeneracy condition.

In this paper we consider the so-called double-phase problem where the phase transition takes place across the interface of the positive and negative phase of minimizers of the functional (Formula presented.) We prove that minimizers exist, are Hölder regular and verify (Formula presented.) in a weak sense. We also prove that their free boundary is (Formula presented.) a.e. with respect to the measure (Formula presented.) whose support is of σ-finite (Formula presented.) -dimensional Hausdorff measure.