This paper explores recent progress related to constraint maps. Building on the exposition in [14], our goal is to provide a clear and accessible account of some of the more intricate arguments behind the main results in this work. Along the way, we include several new results of independent value. In particular, we give optimal geometric conditions on the target manifold that guarantee a unique continuation result for the projected image map. We also prove that the gradient of a minimizing harmonic map (or, more generally, of a minimizing constraint map) is an A∞-weight, and therefore satisfies a strong form of the unique continuation principle. In addition, we outline possible directions for future research and highlight several open problems that may interest researchers working on free boundary problems and harmonic maps.

The primary goal of this paper is to give a precise definition and prove existence and uniqueness of multiphase quadrature domains for subharmonic functions, ensuring that the prescribed measures are supported in the interior of the resulting domains. The approach to proving existence is based on a variational framework, where we minimize an energy functional over so called segregated states. In this respect, we refine earlier results in this direction. However, we also show that this approach alone is not enough for two reasons. First it seems difficult to obtain existence results which ensure that the interior support condition is satisfied. And second it may happen, as we show by an example, that a multiphase quadrature domain exists but is not a minimizer of the energy functional. The main novelty of this work is the study of minimizers, when they exist, of the energy functional over a subset of the segregated states determined by a natural constraint associated with the prescribed measures. From this approach we are able to prove uniqueness, and also give sufficient conditions for existence. We also give an example showing that, unlike the energy minimization and partial balayage approaches which are equivalent in the one-phase case, this equivalence breaks down already in the two-phase setting.

For any subharmonic function u, we prove that |∂_ju| (j=1,…,n) is upper semi-continuous, provided that the super-level sets of u can be touched from the exterior by uniform C^1,Dini domains at every point. This idea extends to a class of general operators, as well as to the boundary behaviour of the gradient of solutions of the Dirichlet problem in a domain whose boundary satisfies this geometric condition.

We study the scattering behavior of an anisotropic inhomogeneous Lipschitz medium at a fixed wave number, continuing our previous work [27] and using free boundary techniques from [32] . Our main results can be categorized into two distinct cases. In the first case, we show that in two dimensions, piecewise C¹ or convex penetrable obstacles with corners, and in higher dimensions, obstacles with edge points, always induce nontrivial scattering for any incoming wave. In the second case, we prove that piecewise C¹ obstacles with corners in two dimensions (and with edge points in higher dimensions) with angles ∉πQ always produce nontrivial scattering for any incoming wave.

In this paper, we study superlinear systems that give rise to free boundaries. Such systems appear for example from the minimization of the energy functional(Formula presented) but solutions can be also understood in an ad hoc viscosity way.First, we prove the optimal regularity of minimizers using a variational approach. Then, we apply a linearization technique to establish the C^1,α-regularity of the “flat” part of the free boundary via a viscosity method. Finally, for minimizing free boundaries, we extend this result to analyticity.

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.

In this expository note, we introduce a vector-valued transmission problem that involves transmission across an unknown (free boundary) set, which is characterized as the preimage of a smooth surface in the target space. Specifically, we explore the question of regularity for solutions to equations where the conductivity exhibits a sharp discontinuity across this unknown set. While this note does not present a complete mathematical analysis, our objective is to outline the model problem and its inherent challenges.

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.

In this survey we will focus on a specific but remarkably ubiquitous family of free boundary problems, those of obstacle-type. We will recount the historical development of the topic together with more recent advances, focusing in particular on vector-valued variational problems with constraints.

We study maps u is an element of W-1,W-2(Omega;M & oline;) that minimize the Alt-Caffarelli energy functional integral(Omega)(divided by Du divided by(2)+q(2)chi(u-1(M))) dx under the condition that the image u(Omega) is confined within M & oline;. Here, Omega denotes a bounded domain in the ambient space R-n (with n >= 1), and M represents a smooth domain in the target space Rm\mathbb{R}<^>mRm (where m >= 2m \ge 2m >= 2). Since our minimizing constraint maps coincide with harmonic maps in the interior of the coincidence set, int(u(-1)(partial derivative M), such maps are prone to developing discontinuities due to their inherent nature. This research marks the commencement of an in-depth analysis of potential singularities that might arise within and around the free boundary. Our first significant contribution is an epsilon-regularity theorem, founded on a novel method of Lipschitz approximation near points exhibiting low energy. Utilizing this approximation and extending the analysis through a bootstrapping approach, we show Lipschitz continuity of our maps whenever the energy is small. Our subsequent key finding reveals that, whenever the complement of MMM is uniformly convex and of class C-3, the maps minimizing the Alt-Caffarelli energy with a positive parameter q exhibit Lipschitz continuity within a universally defined neighborhood of the noncoincidence set u(-1)(M). In particular, this Lipschitz continuity extends to the free boundary. A noteworthy consequence of our findings is the smoothness of flat free boundaries and of the resulting image maps.