Searcharxiv⌕ Search

arXiv subjects

Bozhidar Velichkov

Publications and source records attributed to Bozhidar Velichkov.

At least 37 records · Page 2Linked to original sources
arXiv2024

On the fine structure of the solutions to nonlinear thin two-membrane problems in 2D

We prove a structure theorem for the solutions of nonlinear thin two-membrane problems in dimension two. Using the theory of quasi-conformal maps, we show that the difference of the sheets is topologically equivalent to a solution of the linear thin obstacle problem, thus inheriting its free boundary structure. More precisely, we show that even in the nonlinear case the branching points can only occur in finite number. We apply our methods to one-phase free boundaries approaching a fixed analytic boundary and to the solutions of a one-sided two-phase Bernoulli problem.

math.AP↗
arXiv2024

Boundary regularity of the free interface in spectral optimal partition problems

We consider the problem of optimal partition of a domain with respect to the sum of the principal eigenvalues and we prove for the first time regularity results for the free interface up to fixed boundary. All our results are quantitative and, in particular, we obtain fine estimates on the continuity of the solutions and the oscillation of the free interface (in terms of the modulus of continuity of the normal vector of the fixed boundary), even in the case of domains with low (Dini-type) regularity. Our analysis is based on an Almgren-type monotonicity formula at boundary points and an epiperimetric inequality at points of low frequency, which, together, yield an explicit rate of convergence for blow-up sequences and the boundary strong unique continuation property. Exploiting our quantitative blow-up analysis, we manage to prove clean-up results near one-phase and two-phase points. We define the notion of free interface inside the fixed boundary, and we prove that the subset of points of minimal frequency is regular and that the interior free interface is approaching the boundary orthogonally in a smooth way.

math.AP↗
arXiv2024

The boundary Harnack principle on optimal domains

We give a short and self-contained proof of the Boundary Harnack inequality for a class of domains satisfying some geometric conditions given in terms of a state function that behaves as the distance function to the boundary, is subharmonic inside the domain and satisfies some suitable estimates on the measure of its level sets. We also discuss the applications of this result to some shape optimization and free boundary problems.

math.AP↗
arXiv2023

A capillarity one-phase Bernoulli free boundary problem

We consider a one-phase Bernoulli free boundary problem in a container $D$ - a smooth open subset of $\mathbb{R}^d$ - under the condition that on the fixed boundary $\partial D$ the normal derivative of the solutions is prescribed. We study the regularity of the free boundary (the boundary of the positivity set of the solution) up to $\partial D$ and the structure of the wetting region, which is the contact set between the free boundary and the ($(d-1)$-dimensional) fixed boundary $\partial D$. In particular, we characterize the contact angle in terms of the permeability of the porous container and we show that the boundary of the wetting region is a smooth $(d-2)$-dimensional manifold, up to a (possibly empty) closed set of Hausdorff dimension at most $d-5$.

math.AP↗
arXiv2023

On the dimension of the singular set in optimization problems with measure constraint

In this paper, we prove estimates on the dimension of the singular part of the free boundary for solutions to shape optimization problems with measure constraints. The focus is on the heat conduction problem studied by Aguilera, Caffarelli, and Spruck and the one-phase Bernoulli problem with measure constraint introduced by Aguilera, Alt and Caffarelli. To estimate the Hausdorff dimension of the singular set, we introduce a new formulation of the notion of stability for the one-phase problem along volume-preserving variations, which is preserved under blow-up limits. Finally, the result follows by applying the program developed in [Buttazzo et al. 2022] to this class of domain variation.

math.AP↗
arXiv2023

Regularity for one-phase Bernoulli problems with discontinuous weights and applications

We study a one-phase Bernoulli free boundary problem with weight function admitting a discontinuity along a smooth jump interface. In any dimension $N\ge 2$, we show the $C^{1, α}$ regularity of the free boundary outside of a singular set of Hausdorff dimension at most $N-3$. In particular, we prove that the free boundaries are $C^{1, α}$ regular in dimension $N=2$, while in dimension $N=3$ the singular set can contain at most a finite number of points. We use this result to construct singular free boundaries in dimension $N=2$, which are minimizing for one-phase functionals with weight functions in $L^\infty$ that are arbitrarily close to a positive constant.

math.AP↗
arXiv2023

A one-sided two phase Bernoulli free boundary problem

We study a two-phase free boundary problem in which the two-phases satisfy an impenetrability condition. Precisely, we have two ordered positive functions, which are harmonic in their supports, satisfy a Bernoulli condition on the one-phase part of the free boundary and a two-phase condition on the collapsed part of the free boundary. For this two-membrane type problem, we prove an epsilon-regularity theorem with sharp modulus of continuity. Precisely, we show that at flat points each of the two boundaries is $C^{1,1/2}$ regular surface. Moreover, we show that the remaining singular set has Hausdorff dimension at most $N-5$ as in the case of the classical one-phase problem, $N$ being the dimension of the space.

math.AP↗
arXiv2023

The symmetric (log-)epiperimetric inequality and a decay-growth estimate

We introduce a symmetric (log-)epiperimetric inequality, generalizing the standard epiperimetric inequality, and we show that it implies a growth-decay for the associated energy: as the radius increases energy decays while negative and grows while positive. One can view the symmetric epiperimetric inequality as giving a log-convexity of energy, analogous to the 3-annulus lemma or frequency formula. We establish the symmetric epiperimetric inequality for some free-boundary problems and almost-minimizing currents, and give some applications including a ``propagation of graphicality'' estimate, uniqueness of blow-downs at infinity, and a local Liouville-type theorem.

math.AP↗
arXiv2022

Regularity of the optimal sets for a class of integral shape functionals

We prove {the first} regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain $Ω$ is obtained as the integral of a cost function $j(u,x)$ depending on the solution $u$ of a certain PDE problem on $Ω$. The main feature of these functionals is that the minimality of a domain $Ω$ cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions $j(u,x)=-g(x)u+Q(x)$, where $u$ is the solution of the PDE $-Δu=f$ with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal $u$ from the inwards/outwards optimality of $Ω$ and then we use the stability of $Ω$ with respect to variations with smooth vector fields in order to study the blow-up limits of the state function $u$. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose $\partialΩ$ into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of $\partialΩ$ we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove $C^\infty$ regularity of the regular part of the free boundary when the data are smooth.

math.AP↗
arXiv2021

(Quasi-)conformal methods in two-dimensional free boundary problems

In this paper we study the local behavior of solutions to some free boundary problems. We relate the theory of quasi-conformal maps to the regularity of the solutions to nonlinear thin-obstacle problems; we prove that the contact set is locally a finite union of intervals and we apply this result to the solutions of one-phase Bernoulli problems with geometric constraint. We also introduce a new conformal hodograph transform, which allows to obtain the precise expansion at branch points of both the solutions to the one-phase problem with geometric constraint and a class of symmetric solutions to the two-phase problem, as well as to construct examples of free boundaries with cusp-like singularities.

math.AP↗
arXiv2021

Rectifiability and almost everywhere uniqueness of the blow-up for the vectorial Bernoulli free boundaries

We prove that for minimizers of the vectorial Alt-Caffarelli functional the two-phase singular set of the free boundary is rectifiable and the blow-up is unique almost everywhere on it. While the first conclusion is an application of the recent techniques developed by Naber and Valtorta, the uniqueness part follows from the rectifiability and a new application of the Alt-Caffarelli-Friedman monotonicity formula.

math.AP↗
arXiv2021

Epsilon-regularity for the solutions of a free boundary system

This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions $u$ and $v$, and a domain $Ω$; with $u$ and $v$ being both positive in $Ω$, vanishing simultaneously on $\partialΩ$ and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on $\partialΩ$. Precisely, we consider solutions $u, v \in C(B_1)$ of $$-Δu= f \quad\text{and} \quad-Δv=g\quad\text{in}\quad Ω=\{u>0\}=\{v>0\}\ ,\qquad \frac{\partial u}{\partial n}\frac{\partial v}{\partial n}=Q\quad\text{on}\quad \partialΩ\cap B_1.$$ Our main result is an epsilon-regularity theorem for viscosity solutions of this free boundary system. We prove a partial Harnack inequality near flat points for the couple of auxiliary functions $\sqrt{uv}$ and $\frac12(u+v)$. Then, we use the gained space near the free boundary to transfer the improved flatness to the original solutions. Finally, using the partial Harnack inequality, we obtain an improvement-of-flatness result, which allows to conclude that flatness implies $C^{1,α}$ regularity.

math.AP↗
arXiv2021

Shape optimization problems in control form

We consider a shape optimization problem written in the optimal control form: the governing operator is the $p$-Laplacian in the Euclidean space $\R^d$, the cost is of an integral type, and the control variable is the domain of the state equation. Conditions that guarantee the existence of an optimal domain will be discussed in various situations. It is proved that the optimal domains have a finite perimeter and, under some suitable assumptions, that they are open sets. A crucial difference is between the case $p>d$, where the existence occurs under very mild conditions, and the case $p\le d$, where additional assumptions have to be made on the data.

math.OC↗
arXiv2020

A two-phase problem with Robin conditions on the free boundary

We study for the first time a two-phase free boundary problem in which the solution satisfies a Robin boundary condition. We consider the case in which the solution is continuous across the free boundary and we prove an existence and a regularity result for minimizers of the associated variational problem. Finally, in the appendix, we give an example of a class of Steiner symmetric minimizers.

math.AP↗
arXiv2020

Regularity of the optimal sets for the second Dirichlet eigenvalue

This paper is dedicated to the regularity of the optimal sets for the second eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set $Ω$ minimizes the functional \[ \mathcal F_Λ(Ω)=λ_2(Ω)+Λ|Ω|, \] among all subsets of a smooth bounded open set $D\subset \mathbb{R}^d$, where $λ_2(Ω)$ is the second eigenvalue of the Dirichlet Laplacian on $Ω$ and $Λ>0$ is a fixed constant, then $Ω$ is equivalent to the union of two disjoint open sets $Ω_+$ and $Ω_-$, which are $C^{1,α}$-regular up to a (possibly empty) closed set of Hausdorff dimension at most $d-5$, contained in the one-phase free boundaries $D\cap \partialΩ_+\setminus\partialΩ_-$ and $D\cap\partialΩ_-\setminus\partialΩ_+$.

math.AP↗
arXiv2020

On the logarithmic epiperimetric inequality for the obstacle problem

We give three different proofs of the log-epiperimetric inequality at singular points for the obstacle problem. In the first, direct proof, we write the competitor explicitly; the second proof is also constructive, but this time the competitor is given through the solution of an evolution problem on the sphere. We compare the competitors obtained in the different proofs and their relation to other similar results that appeared recently. Finally, in the appendix, we give a general theorem, which can be applied also in other contexts and in which the construction of the competitor is reduced to finding a flow satisfying two differential inequalities.

math.AP↗