Some free boundary problems recast as nonlocal parabolic equations

Existence and uniqueness of classical solutions for the multidimensional expanding Hele{Shaw problem are proved.

In this paper, we prove the existence of smooth solutions in Sobolev spaces to fully nonlinear and nonlocal parabolic equations with critical index. Our argument is to transform the fully nonlinear equation into a quasi-linear nonlocal parabolic equation.

The field of the mathematical and numerical analysis of systems of nonlinear pdes involving interfaces and free boundaries is a burgeoning area of research. Many such systems arise from mathematical models in ma- terial science and fluid dynamics such as phase separation in alloys, crystal growth, dynamics of multiphase fluids and epitaxial growth. In applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimiza- tion problems with pde constraints including free boundaries. It is now timely to consider such control problems because of the maturity of the field of com- putational free boundary problems. The aim of the mini-workshop was to bring together leading experts and young researchers from the separate fields of numerical free boundary problems and optimal control in order to estab- lish links and to identify suitable model problems to serve as paradigms for progressing knowledge of optimal control of free boundaries.

formula omitted]-maximal regularity of nonlocal parabolic equations and applications

We study the following two phase elliptic singular perturbation problem: Δ u ε = β ε ( u ε ) + f ε , in Ω ⊂ R N , where ε > 0 , β ε ( s ) = 1 ε β ( s ε ) , with β a Lipschitz function satisfying β > 0 in ( 0 , 1 ) , β ≡ 0 outside ( 0 , 1 ) and ∫ β ( s ) d s = M . The functions u ε and f ε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present. We obtain uniform estimates, we pass to the limit ( ε → 0 ) and we show that limit functions are solutions to the two phase free boundary problem: Δ u = f χ { u ≢ 0 } in Ω ∖ ∂ { u > 0 } , | ∇ u + | 2 − | ∇ u − | 2 = 2 M on Ω ∩ ∂ { u > 0 } , where f = lim f ε , in a viscosity sense and in a pointwise sense at regular free boundary points. In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions. Some of the results obtained are new even in the case f ε ≡ 0 . The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary.

Uniform Lipschitz regularity for classes of minimizers in two phase free boundary problems in Orlicz spaces with small density on the negative phase

In this article, we have established existence of a solution to the 2 -phase free boundary problem for some fully nonlinear elliptic equations and also shown the free boundary has finite Hn−1 Hausdorff measure and a normal in a measuretheoretic sense Hn−1 almost everywhere. The regularity theory developed in [9] and [10] for this free boundary problem then leads to the fact that the free boundary is locally a C1,α surface near Hn−1-a.e. point.

One-dimensional Shape Memory Alloy Problem with Duhem Type of Hysteresis Operator.- Existence and Uniqueness Results for Quasi-linear Elliptic and Parabolic Equations with Nonlinear Boundary Conditions.- Finite Time Localized Solutions of Fluid Problems with Anisotropic Dissipation.- Parabolic Equations with Anisotropic Nonstandard Growth Conditions.- Parabolic Systems with the Unknown Dependent Constraints Arising in Phase Transitions.- The N-membranes Problem with Neumann Type Boundary Condition.- Modelling, Analysis and Simulation of Bioreactive Multicomponent Transport.- Asymptotic Properties of the Nitzberg-Mumford Variational Model for Segmentation with Depth.- The ?-Laplacian First Eigenvalue Problem.- Comparison of Two Algorithms to Solve the Fixed-strike Amerasian Options Pricing Problem.- Nonlinear Diffusion Models for Self-gravitating Particles.- Existence, Uniqueness and an Explicit Solution for a One-Phase Stefan Problem for a Non-classical Heat Equation.- Dislocation Dynamics: a Non-local Moving Boundary.- Bermudean Approximation of the Free Boundary Associated with an American Option.- Steady-state Bingham Flow with Temperature Dependent Nonlocal Parameters and Friction.- Some P.D.E.s with Hysteresis.- Embedding Theorem for Phase Field Equation with Convection.- A Dynamic Boundary Value Problem Arising in the Ecology of Mangroves.- Wave Breaking over Sloping Beaches Using a Coupled Boundary Integral-Level Set Method.- Finite Difference Schemes for Incompressible Flows on Fully Adaptive Grids.- Global Solvability of Constrained Singular Diffusion Equation Associated with Essential Variation.- Capillary Mediated Melting of Ellipsoidal Needle Crystals.- Boundary Regularity at {t = 0} for a Singular Free Boundary Problem.- Fast Reaction Limits and Liesegang Bands.- Numerical Modeling of Surfactant Effects in Interfacial Fluid Dynamics.- The Value of an American Basket Call with Dividends Increases with the Basket Volatility.- Mathematical Modelling of Nutrient-limited Tissue Growth.- Asymptotic Hysteresis Patterns in a Phase Separation Problem.- Obstacle Problems for Monotone Operators with Measure Data.- Piecewise Constant Level Set Method for Interface Problems.- Dynamics of a Moving Reaction Interface in a Concrete Wall.- Adaptive Finite Elements with High Aspect Ratio for Dendritic Growth of a Binary Alloy Including Fluid Flow Induced by Shrinkage.- A Free Boundary Problem for Nonlocal Damage Propagation in Diatomites.- Concentrating Solutions for a Two-dimensional Elliptic Problem with Large Exponent in Nonlinearity.- Existence of Weak Solutions for the Mullins-Sekerka Flow.- Existence and Approximation Results for General Rate-independent Problems via a Variable Time-step Discretization Scheme.- Global Attractors for the Quasistationary Phase Field Model: a Gradient Flow Approach.- Aleksandrov and Kelvin Reflection and the Regularity of Free Boundaries.- Solvability for a PDE Model of Regional Economic Trend.- Surface Energies in Multi-phase Systems with Diffuse Phase Boundaries.- High-order Techniques for Calculating Surface Tension Forces.- Simulation of a Model of Tumors with Virus-therapy.- Asymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction.

The regularity properties of nonlocal fractional elliptic and parabolic equations in vector-valued Besov spaces are studied. The uniform B p , q s B_{p,q}^{s} -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. In particular, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, the maximal regularity properties of the nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional parabolic equations are studied.

We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [J. Cao, Y. Du, F. Li and W.-T. Li, The dynamics of a Fisher–KPP nonlocal diffusion model with free boundaries, J. Functional Anal. 277 (2019) 2772–2814; C. Cortazar, F. Quiros and N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound. 21 (2019) 441–462]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form [Formula: see text] for small [Formula: see text], where [Formula: see text] has compact support. We also give an estimate of the error term of the approximation by some positive power of [Formula: see text].

Lagrangian coordinates in free boundary problems for parabolic equations

In this paper, we consider the symmetry properties of positive solutions for nonlocal parabolic equations in the whole space. We obtain various asymptotic maximum principles for carrying out the asymptotic method of moving planes. With the help of these results, we show that if the equation converges to a symmetric one, then the solutions will converge to radially symmetric functions. The methods and techniques used here can be easily applied to study a variety of nonlocal parabolic equations with more general operators and nonlinear terms.

We consider the spreading, driven by surface tension, of a thin liquid droplet on a plane solid surface. In lubrication approximation, this phenomenon may be modeled by a class of free boundary problems for fourth order nonlinear degenerate parabolic equations, the free boundary being defined as the contact line where liquid, solid and vapor meet. Our interest is on an effective free boundary condition which has been recently proposed by Ren and E: it includes into the model the effect of frictional forces at the contact line, which arises from the deviation of the contact angle from its equilibrium value. In this note we outline the lubrication approximation of this condition, we describe the dissipative structure and the traveling wave profiles of the resulting free boundary problem, and we prove existence and uniqueness of a class of traveling wave solutions which naturally emerges from the formal asymptotic analysis.

We use the method of moving planes to prove the radial symmetry and monotonicity of solutions of fractional parabolic equations in the unit ball. Since the fractional Laplacian operator is a linear operator, we investigate the maximal regularity of nonlocal parabolic fractional Laplacian equations in the unit ball. The maximal regularity of nonlocal parabolic fractional Laplacian equations guarantees the existence of solutions in the unit ball. Based on these conditions, we first establish a maximum principle in a parabolic cylinder, and the principles provide a starting position to apply the method of moving planes. Then, we consider the fractional parabolic equations and derive the radial symmetry and monotonicity of solutions in the unit ball.

In this paper, we demonstrate that the so-called expansion of positivity holds true for doubly nonlinear nonlocal parabolic type equations, having the fractional p-Laplace type operator and a power-nonlinearity in the time-derivative. The exponential time-stretching method originally developed for the local equations is transparently extended to the nonlocal equations in the scaling regime intrinsic to the doubly nonlinear nonlocal parabolic operator. The nonlocal effect of the nonlocal equations are given by the so-called tail.