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Ivan Blank

Publications and source records attributed to Ivan Blank.

9 recordsLinked to original sources
arXiv2019

The geometry of the triple junction between three fluids in equilibrium

We conduct an analysis of the blow up at the triple junction of three fluids in equilibrium. Energy minimizers have been shown to exist in the class of functions of bounded variations, and the classical theory implies that an interface between two fluids is an analytic surface. We prove two monotonicity formulas at the triple junction for the three-fluid configuration, and show that blow up limits exist and are always cones. We discuss some of the geometric consequences of our results.

math.AP↗
arXiv2019

A Uniqueness Theorem for Mean Value Sets for Elliptic Divergence Form Operators

We give background which shows the connection between the mean value theorem and the obstacle problem, and then we prove that a set is a mean value set for an elliptic operator of the form $Lu := \partial_i (a^{ij}(x) \partial_j u(x))$ if and only if it arises as the noncontact set of an obstacle problem involving the Green's function of the operator.

math.AP↗
arXiv2018

Nondegenerate Motion of Singular Points in Obstacle Problems with Varying Data

Recent work by Serfaty and Serra give a formula for the velocity of the free boundary of the obstacle problem at regular points [Serfaty-Serra 2018], and much older work by King, Lacey, and Vazquez gives an example of a singular free boundary point (in the Hele-Shaw flow) that remains stationary for a positive amount of time [King-Lacey-Vazquez 1995]. The authors show how singular free boundaries in the obstacle problem in some settings move immediately in response to varying data. Three applications of this result are given, and in particular, the authors show a uniqueness result: For sufficiently smooth elliptic divergence form operators on domains in $\mathrm{I \! R}^n$ and for the Laplace-Beltrami operator on a smooth manifold, the boundaries of distinct mean value sets (of the type found in [Blank-Hao 2015] and [Benson-Blank-LeCrone 2018]) which are centered at the same point do not intersect.

math.AP↗
arXiv2018

Perturbed Obstacle Problems in Lipschitz Domains: Linear Stability and Non-degeneracy in Measure

We consider the classical obstacle problem on bounded, connected Lipschitz domains $D \subset \mathbb{R}^n$. We derive quantitative bounds on the changes to contact sets under general perturbations to both the right hand side and the boundary data for obstacle problems. In particular, we show that the Lebesgue measure of the symmetric difference between two contact sets is linearly comparable to the $L^1$-norm of perturbations in the data.

math.AP↗
arXiv2017

Geometry of mean value sets for general divergence form uniformly elliptic operators

In the Fermi Lectures on the obstacle problem in 1998, Caffarelli gave a proof of the mean value theorem which extends to general divergence form uniformly elliptic operators. In the general setting, the result shows that for any such operator $L$ and at any point $x_0$ in the domain, there exists a nested family of sets $\{ D_r(x_0) \}$ where the average over any of those sets is related to the value of the function at $x_0.$ Although it is known that the $\{ D_r(x_0) \}$ are nested and are comparable to balls in the sense that there exists $c, C$ depending only on $L$ such that $B_{cr}(x_0) \subset D_r(x_0) \subset B_{Cr}(x_0)$ for all $r > 0$ and $x_0$ in the domain, otherwise their geometric and topological properties are largely unknown. In this paper we begin the study of these topics and we prove a few results about the geometry of these sets and give a couple of applications of the theorems.

math.AP↗
arXiv2014

The Mean Value Theorem and Basic Properties of the Obstacle Problem for Divergence Form Elliptic Operators

In 1963, Littman, Stampacchia, and Weinberger proved a mean value theorem for elliptic operators in divergence form with bounded measurable coefficients. In the Fermi lectures in 1998, Caffarelli stated a much simpler mean value theorem for the same situation, but did not include the details of the proof. We show all of the nontrivial details needed to prove the formula stated by Caffarelli, and in the course of showing these details we establish some of the basic facts about the obstacle problem for general elliptic divergence form operators, in particular, we show a basic quadratic nondegeneracy property.

math.AP↗
arXiv2013

Reifenberg Flatness of Free Boundaries in Obstacle Problems with VMO Ingredients

We study the obstacle problem with an elliptic operator in divergence form. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary in the case where the coefficients are in VMO.

math.AP↗
arXiv2013

The Caffarelli Alternative in Measure for the Nondivergence Form Elliptic Obstacle Problem with Principal Coefficients in VMO

We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper "The regularity of free boundaries in higher dimensions."

math.AP↗