arXiv subjects

Bozhidar Velichkov

Publications and source records attributed to Bozhidar Velichkov.

At least 19 recordsLinked to original sources

arXiv2026

An elliptic regularization approach to the Stefan problem

In this paper, we develop the theory for the two-phase Stefan problem with finite energy, possibly non-empty mushy region, and space-dependent melting temperature. Specifically, we prove the existence of weak solutions with an elliptic regularization scheme. Our existence theorem provides information about the regularity of the solutions: we prove that the temperature of weak solutions is in $H^1$ for all times, that the enthalpy is well defined and bounded for all times, and that both the enthalpy and the temperature are weakly continuous in time. Finally, we establish a comparison principle for weak solutions on general unbounded domains and use it to show that every weak solution is recovered by the approximation scheme.

math.AP↗

arXiv2026

On a Multiphase Vectorial Bernoulli Free Boundary Problem

We study the regularity of minimizers of a multiphase vectorial Bernoulli free boundary problem. This problem consists in a minimization problem for the Bernoulli functional over families of Sobolev functions with disjoint supports and non trivial grouping. We prove that minimizers exist, are locally Lipschitz continuous, and that their free boundaries do not contain points where three or more phases meet. Our main regularity result establishes that the free boundary is locally a $C^{1,\eta}$ graph near two-phase and branching points for some $\eta >0$.

math.AP↗

arXiv2026

Fine structure of the two-phase Bernoulli free boundaries in 2D

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 new geometric two-phase problem for capillary minimal surfaces. We also apply this method to the obstacle problem establishing a connection between the directional derivatives of solutions to the obstacle problem and the linear thin two-membrane problem.

math.AP↗

arXiv2026

On the blow-up of the vectorial Bernoulli free boundary problem

In this paper, we complete the classification of the blow-up limits of minimizers of the vectorial Bernoulli free boundary problem. Furthermore, we study the vectorial Bernoulli free boundary problem in a bounded box $D$, with a constraint $m$ on the measure of the positivity set, and the asymptotic of minimizers as the measure constraint $m$ tends to $|D|$. Such a study with a linear datum on the fixed boundary is the main ingredient for the characterization of the singular homogeneous global solutions of the vectorial problem and, thus, for the classification of the blow-up limits.

math.AP↗

arXiv2025

Vectorial Bernoulli Problems and Free Boundary Systems

In this survey we go through some of the recent results about the regularity of vectorial free boundary problems of Bernoulli type and free boundary systems. The aim is to illustrate the general methodologies as well as to outline a selection of notable open questions.

math.AP↗

arXiv2025

A survey on the optimal partition problem

This survey synthesizes the current state of the art on the regularity theory for solutions to the optimal partition problem. Namely, we consider non-negative, vector-valued Sobolev functions whose components have mutually disjoint support, and which are either local minimizers of the Dirichlet energy or, more generally, critical points satisfying a system of variational inequalities. This is particularly meaningful as the problem has emerged on several occasions and in diverse contexts: our aim is then to provide a coherent point of view and an up-to-date account of the progress concerning regularity of the solutions and their free boundaries, both in the interior and up to a fixed boundary.

math.AP↗

arXiv2025

Regularity for free boundary surfaces minimizing degenerate area functionals

We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to $d(\cdot,\mathbb{R}^n\setminus\Omega)^a$ near $\partial\Omega$ and $a>0$. This implies that the boundaries of almost-minimizers are $C^{1,\gamma_0}$-surfaces that touch $\partial \Omega$ orthogonally, up to a Singular Set $\mathrm{Sing}(\partial E)$ whose Hausdorff dimension satisfies the bound $d_{\mathscr{H}}(\mathrm{Sing}(\partial E)) \leq n +a -(5+\sqrt{8})$.

math.AP↗

arXiv2025

Some remarks on singular capillary cones with free boundary

We study minimizing singular cones with free boundary associated with the capillarity problem. Precisely, we provide a stability criterion $\`a$ la Jerison-Savin for capillary hypersurfaces and show that, in dimensions up to $4$, minimizing cones with non-sign-changing mean curvature are flat. We apply this criterion to minimizing capillary drops and, additionally, establish the instability of non-trivial axially symmetric cones in dimensions up to $6$. The main results are based on a Simons-type inequality for a class of convex, homogeneous, symmetric functions of the principal curvatures, combined with a boundary condition specific to the capillary setting.

math.DG↗

arXiv2025

Existence and regularity in the fully nonlinear one-phase free boundary problem

We consider viscosity solution to one-phase free boundary problems for general fully nonlinear operators and free boundary condition depending on the normal vector. We show existence of viscosity solutions via the Perron's method and we prove $C^{2,\alpha}$ regularity of flat free boundaries via a quadratic improvement of flatness. Finally, we obtain the higher regularity of the free boundary via an hodograph transform.

math.AP↗

arXiv2024

Structure of the free interfaces near triple junction singularities in harmonic maps and optimal partition problems

We consider energy-minimizing harmonic maps into trees and we prove the regularity of the singular part of the free interface near triple junction points. Precisely, by proving a new epiperimetric inequality, we show that around any point of frequency $3/2$, the free interface is composed of three $C^{1,\alpha}$-smooth $(d-1)$-dimensional manifolds (composed of points of frequency $1$) with common $C^{1,\alpha}$-regular boundary (made of points of frequency $3/2$) that meet along this boundary at 120 degree angles. Our results also apply to spectral optimal partition problems for the Dirichlet eigenvalues.

math.AP↗

arXiv2024

An epiperimetric inequality for odd frequencies in the thin obstacle problem

We prove for the first time an epiperimetric inequality for the thin obstacle Weiss' energy with odd frequencies and we apply it to solutions to the thin obstacle problem with general $C^{k,\gamma}$. In particular, we obtain the rate of convergence of the blow-up sequences at points of odd frequencies and the regularity of the strata of the corresponding contact set. We also recover the frequency gap for odd frequencies obtained by Savin and Yu.

math.AP↗

arXiv2024

Unique continuation for nonlinear variational problems

This paper is dedicated to the unique continuation properties of the solutions to nonlinear variational problems. Our analysis covers the case of nonlinear autonomous functionals depending on the gradient, as well as more general double phase and multiphase functionals with $(2,q)$-growth in the gradient. We show that all these cases fall in a class of nonlinear functionals for which we are able to prove weak and strong unique continuation via the almost-monotonicity of Almgren's frequency formula. As a consequence, we obtain estimates on the dimension of the set of points at which both the solution and its gradient vanish.

math.AP↗

arXiv2024

On the boundary branching set of the one-phase problem

We consider minimizers of the one-phase Bernoulli free boundary problem in domains with analytic fixed boundary. In any dimension $d$, we prove that the branching set at the boundary has Hausdorff dimension at most $d-2$. As a consequence, we also obtain an analogous estimate on the branching set for solutions to the two-phase problem under an analytic separation condition. Moreover, as a byproduct of our analysis we obtain strong boundary unique continuation results for quasilinear operators and thin-obstacle variational inequalities. The approach we use is based on the (almost-)monotonicity of a boundary Almgren-type frequency function, obtained via regularity estimates and a Calder\'on-Zygmund decomposition in the spirit of Almgren-De Lellis-Spadaro.

math.AP↗

arXiv2024

Classical solutions to the soap film capillarity problem for plane boundaries

We study the soap film capillarity problem, in which soap films are modeled as sets of least perimeter among those having prescribed (small) volume and satisfying a topological spanning condition. When the given boundary is the closed tubular neighborhood in $\mathbb{R}^3$ of a smooth Jordan curve (or, more generally, the closed tubular neighborhood in $\mathbb{R}^d$ of a smooth embedding of $\mathbb{S}^{d-2}$ in a hyperplane), we prove existence and uniqueness of classical minimizers, for which the collapsing phenomenon does not occur. We show that the boundary of the unique minimizer is the union of two symmetric smooth normal graphs over a portion of the plane; the graphs have positive constant mean curvature bounded linearly in terms of the volume parameter, and meet the boundary of the tubular neighbourhood orthogonally. Moreover, we prove uniform bounds on the sectional curvatures in order to show that the boundaries of solutions corresponding to varying volumes are ordered monotonically and produce a foliation of space by constant mean curvature hypersurfaces.

math.AP↗

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↗