scholarly journals Two-Phase Free Boundary Problems: From Existence to Smoothness

2017 ◽  
Vol 17 (2) ◽  
Author(s):  
Daniela De Silva ◽  
Fausto Ferrari ◽  
Sandro Salsa
Keyword(s):  
Free Boundary ◽  
Viscosity Solutions ◽  
Free Boundary Problems ◽  
Elliptic Operators ◽  
Free Boundaries ◽  
Two Phase ◽  
Boundary Problems ◽  
Open Questions

AbstractWe describe the theory we developed in recent times concerning two-phase free boundary problems governed by elliptic operators with forcing terms. Our results range from existence of viscosity solutions to smoothness of both solutions and free boundaries. We also discuss some open questions, possible object of future investigation.

scholarly journals Lipschitz Regularity of Solutions to Two-phase Free Boundary Problems

2017 ◽  
Vol 2019 (7) ◽  
pp. 2204-2222 ◽  
Author(s):  
D De Silva ◽  
O Savin
Keyword(s):  
Free Boundary ◽  
Viscosity Solutions ◽  
Lipschitz Continuity ◽  
Free Boundary Problems ◽  
Linear Operators ◽  
Regularity Of Solutions ◽  
Two Phase ◽  
Boundary Problems ◽  
Lipschitz Regularity ◽  
Non Linear

AbstractWe prove Lipschitz continuity of viscosity solutions to a class of two-phase free boundary problems governed by fully non-linear operators.

scholarly journals On the Two-phase Fractional Stefan Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 437-458 ◽  
Author(s):  
Félix del Teso ◽  
Jørgen Endal ◽  
Juan Luis Vázquez
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Free Boundary Problems ◽  
Classical Problem ◽  
Nonlocal Diffusion ◽  
Two Phase ◽  
Boundary Problems ◽  
Numerical Studies ◽  
The One ◽  
Evolution Type

AbstractThe classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

scholarly journals Free boundaries in problems with hysteresis

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140271 ◽  
Author(s):  
D. E. Apushkinskaya ◽  
N. N. Uraltseva
Keyword(s):  
Free Boundary ◽  
Sobolev Class ◽  
Free Boundary Problems ◽  
Strong Solutions ◽  
Chemical Processes ◽  
Free Boundaries ◽  
Open Problems ◽  
Boundary Problems ◽  
Hysteresis Operator ◽  
With Memory

Here, we present a survey concerning parabolic free boundary problems involving a discontinuous hysteresis operator. Such problems describe biological and chemical processes ‘with memory’ in which various substances interact according to hysteresis law. Our main objective is to discuss the structure of the free boundaries and the properties of the so-called ‘strong solutions’ belonging to the anisotropic Sobolev class with sufficiently large q . Several open problems in this direction are proposed as well.

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