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Ovidiu Savin

Publications and source records attributed to Ovidiu Savin.

At least 37 records · Page 2Linked to original sources
arXiv2022

Uniform density estimates and $Γ$-convergence for the Alt-Phillips functional of negative powers

We obtain density estimates for the free boundaries of minimizers $u \ge 0$ of the Alt-Phillips functional involving negative power potentials $$\int_Ω\left(|\nabla u|^2 + u^{-γ} χ_{\{u>0\}}\right) \, dx, \quad \quad γ\in (0,2).$$ These estimates remain uniform as the parameter $γ\to 2$. As a consequence we establish the uniform convergence of the corresponding free boundaries to a minimal surface as $γ\to 2$. The results are based on the $Γ$-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional $$\int_Ω|\nabla u|^2 dx + Per_Ω(\{ u=0\}),$$ considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.

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arXiv2022

The Alt-Phillips functional for negative powers

We develop the free boundary regularity for nonnegative minimizers of the Alt-Phillips functional for negative power potentials $$\int_Ω\left(\frac 1 2 |\nabla u|^2 + u^γ χ_{\{u>0\}}\right) \, dx, \quad \quad γ\in (-2,0),$$ and establish a $Γ$-convergence result of the rescaled energies to the perimeter functional as $γ\to -2$.

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arXiv2021

Half-space solutions with $7/2$ frequency in the thin obstacle problem

For the thin obstacle problem in 3d, we show that half-space solutions form an isolated family in the space of $7/2$-homogeneous solutions. For a general solution with one blow-up profile in this family, we establish the rate of convergence to this profile. As a consequence, we obtain regularity of the free boundary near such contact points.

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arXiv2021

Free boundary regularity in the triple membrane problem

We investigate the regularity of the free boundaries in the 3 elastic membranes problem. We show that the two free boundaries corresponding to the coincidence regions between consecutive membranes are $C^{1,\log}$-hypersurfaces near a regular intersection point. We also study two types of singular intersections. The first type of singular points are locally covered by a $C^{1,α}$-hypersurface. The second type of singular points stratify and each stratum is locally covered by a $C^1$-manifold.

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arXiv2021

On the Parabolic Boundary Harnack Principle

We investigate the parabolic Boundary Harnack Principle for both divergence and non-divergence type operators by the analytical methods we developed in the elliptic context. Besides the classical case, we deal with less regular space-time domains, including slit domains.

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arXiv2021

On divergent fractional Laplace equations

We consider the divergent fractional Laplace operator presented in [Dipierro-Savin-Valdinoci, Rev. Mat. Iberoam.] and we prove three types of results. Firstly, we show that any given function can be locally shadowed by a solution of a divergent fractional Laplace equation which is also prescribed in a neighborhood of infinity. Secondly, we take into account the Dirichlet problem for the divergent fractional Laplace equation, proving the existence of a solution and characterizing its multiplicity. Finally, we take into account the case of nonlinear equations, obtaining a new approximation results. These results maintain their interest also in the case of functions for which the fractional Laplacian can be defined in the usual sense.

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arXiv2021

On the fine regularity of the singular set in the nonlinear obstacle problem

We revisit and sharpen the results from our previous work, where we investigated the regularity of the singular set of the free boundary in the nonlinear obstacle problem. As in the work of Figalli-Serra on the classical obstacle problem, we show that each stratum can be further decomposed into a `good' part and an `anomalous' part, where the former is covered by $C^{1,1-}$ manifolds, and the latter is of lower dimension.

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arXiv2020

Perturbative estimates for the one-phase Stefan Problem

We provide perturbative estimates for the one-phase Stefan free boundary problem and obtain the regularity of flat free boundaries via a linearization technique in the spirit of the elliptic counterpart established by the first author.

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arXiv2020

Regularity of the singular set in the fully nonlinear obstacle problem

For the obstacle problem involving a convex fully nonlinear elliptic operator, we show that the singular set in the free boundary stratifies. The top stratum is locally covered by a $C^{1,α}$-manifold, and the lower strata are covered by $C^{1,\log^\varepsilon}$-manifolds. This essentially recovers the regularity result obtained by Figalli-Serra when the operator is the Laplacian.

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arXiv2020

Nonlocal minimal graphs in the plane are generically sticky

We prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities. More precisely, we show that if a nonlocal minimal graph in a slab is continuous up to the boundary, then arbitrarily small perturbations of the far-away data produce boundary discontinuities. Hence, either a nonlocal minimal graph is discontinuous at the boundary, or a small perturbation of the prescribed conditions produces boundary discontinuities. The proof relies on a sliding method combined with a fine boundary regularity analysis, based on a discontinuity/smoothness alternative. Namely, we establish that nonlocal minimal graphs are either discontinuous at the boundary or their derivative is Hölder continuous up to the boundary. In this spirit, we prove that the boundary regularity of nonlocal minimal graphs in the plane "jumps" from discontinuous to $C^{1,γ}$, with no intermediate possibilities allowed. In particular, we deduce that the nonlocal curvature equation is always satisfied up to the boundary. As an interesting byproduct of our analysis, one obtains a detailed understanding of the "switch" between the regime of continuous (and hence differentiable) nonlocal minimal graphs to that of discontinuous (and hence with differentiable inverse) ones.

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arXiv2019

On certain degenerate one-phase free boundary problems

We develop an existence and regularity theory for a class of degenerate one-phase free boundary problems. In this way we unify the basic theories in free boundary problems like the classical one-phase problem, the obstacle problem, or more generally for minimizers of the Alt-Phillips functional.

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arXiv2019

A short proof of Boundary Harnack Inequality

We give a direct analytic proof of the classical Boundary Harnack inequality for solutions to linear uniformly elliptic equations in either divergence or non-divergence form.

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arXiv2019

Boundary properties of fractional objects: flexibility of linear equations and rigidity of minimal graphs

The main goal of this article is to understand the trace properties of nonlocal minimal graphs in~$\R^3$, i.e. nonlocal minimal surfaces with a graphical structure. We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum. This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can be prescribed arbitrarily, up to a small error. We remark that, at a formal level, the linearization of the trace of a nonlocal minimal graph is given by the fractional normal derivative of a fractional Laplace problem, therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type present very specific properties which are strikingly different from those of other problems of fractional type which are apparently similar, but diverse in structure, and the nonlinear case given by the nonlocal minimal graphs turns out to be significantly more rigid than its linear counterpart.

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arXiv2019

Almost minimizers of the one-phase free boundary problem

We consider almost minimizers to the one-phase energy functional and we prove their optimal Lipschitz regularity and partial regularity of their free boundary. These results were recently obtained by David and Toro, and David, Engelstein, and Toro. Our proofs provide a different method based on a non-infinitesimal notion of viscosity solutions.

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