Correction to: Free Boundary Problem for a Gas Bubble in a Liquid, and Exponential Stability of the Manifold of Spherically Symmetric Equilibria

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

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.

Deep learning of free boundary and Stefan problems

We provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting wavetrains is close to the uniquely determined exact solution for small wavelengths (ε). Waves reflecting off of fixed noncharacteristic boundaries and off of multidimensional shocks are considered under the assumption that the underlying fixed (respectively, free) boundary problem is uniformly spectrally stable in the sense of Kreiss (respectively, Majda). Our results apply to a general class of problems that includes the compressible Euler equations; as a corollary we rigorously justify the leading term in the geometric optics expansion of highly oscillatory multidimensional shock solutions of the Euler equations. An earlier stability result of this type [21] was obtained by a method that required the construction of high-order approximate solutions. That construction in turn was possible only under a generically valid (absence of) small divisors assumption. Here we are able to remove that assumption and avoid the need for high-order expansions by studying associated singular (because they involve coefficients of order ) fixed and free boundary problems. The analysis applies equally to systems that cannot be written in conservative form.

In a couple of recent papers free boundary problems and a class of conformal mappings involving curvilinear quadrilaterals were analysed primarily using transform methods. In both the free boundary and conformal mapping problems there is an underlying relation with Fuchsian differential equations and an alternative, conceptually simpler, solution technique for these problems would use solutions of these Fuchsian equations. Thus it was conjectured in those papers that a class of Fuchsian differential equations, commonly known as Heun's equation, has in some special, but relatively important, cases degenerate solutions; these involve hypergeometric functions. This complementary solution method is, in many cases, more convenient than the transform approach. The purpose of this paper is to explore the connections with the Fuchsian equations directly, and use the solutions to solve free boundary problems.The specific examples treated here come from a quasi–steady approximation to solidification problems and are not without interest in their own right. There are few analytical solutions for solidification problems in geometries of practical interest, and the solutions found here should be of use in that regard.

Free boundary problems deal with solving partial differential equations in a domain, a part of whose boundary is unknown – the so-called free boundary. Beside the standard boundary conditions that are needed in order to solve the partial differential equation, an additional boundary condition is imposed at the free boundary. One aims thus to determine both, the free boundary and the solution of the partial differential equation. This thesis is dedicated to the solution of the generalized exterior Bernoulli free boundary problem which is an important model problem for developing algorithms in a broad band of applications such as optimal design, fluid dynamics, electromagnentic shaping etc. Due to its various advantages in the analysis and implementation, the trial method, which is a fixed-point type iteration method, has been chosen as numerical method. The iterative scheme starts with an initial guess of the free boundary. Given one boundary condition at the free boundary, the boundary element method is applied to compute an approximation of the violated boundary data. The free boundary is then updated such that the violated boundary condition is satisfied at the new boundary. Taylor’s expansion of the violated boundary data around the actual boundary yields the underlying equation, which is formulated as an optimization problem for the sought update function. When a target tolerance is achieved the iterative procedure stops and the approximate solution of the free boundary problem is detected. How efficient or quick the trial method is converging depends significantly on the update rule for the free boundary, and thus on the violated boundary condition. Firstly, the trial method with violated Dirichlet data is examined and updates based on the first and the second order Taylor expansion are performed. A thorough analysis of the convergence of the trial method in combination with results from shape sensitivity analysis motivates the development of higher order convergent versions of the trial method. Finally, the gained experience is exploited to draw very important conclusions about the trial method with violated Neumann data, which is until now poorly explored and has never been numerically implemented.

This paper studies conditions for local (in time) solvability of a qualitatively new singularllimit problem, the free (unknown) boundary problem appearing recently. In fact, there are not so many different free boundary problems, which corresponds to not so large a variety of principally different phase transitions of the first and second kinds. Therefore, the appearance of principally new problems elicits interest. This paper studies structural features of a certain problem on the basis of a certain method developed previously, precisely, the localization method [1, 3, 9].

We derive a convex relaxation principle for a large class of non convex variational problems where the functional to be minimized involves a one homogeneous gradient energy. This applies directly to free boundary or multiphase problems in the case of the classical total variation or of some anisotropic variants. The underlying argument is an exclusion principle which states that any global minimizer avoids taking values in the intervals where the lower order potential is nonconvex. This allows using duality methods and deriving a saddle point characterization of the global minimizers. A numerical validation of our principle is presented in the case of several free boundary and multiphase problems that we treat through a primal-dual algorithm. The accuracy of the interfaces and the convergence of the algoritm benefit in a large way of a new epigraphical projection method that we introduced to tackle the non differentiability of the convexified Lagrangian.

Previous article Next article Analysis of a Model of Percolation in a Gently Sloping Sand-BankCharles M. Elliott and Avner FriedmanCharles M. Elliott and Avner Friedmanhttps://doi.org/10.1137/0516071PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA free boundary problem associated with the percolation of sea-water in a “nearly” flat sand-bank is considered. The wet and dry zones of the beach are studied.[1] J. M. Aitchison, , C. M. Elliott and , J. R. Ockendon, Percolation in gently sloping beaches, IMA J. Appl. Math., 30 (1983), 269–287 84i:76067 0536.76085 CrossrefISIGoogle Scholar[2] J. M. Aitchison, , A. G. Newlands and , C. P. Please, Percolation in intertidal sand-bands-a constrained harmonic problem, CEGB Research Rept., TPRD/L/2344/N82, CERL, Kelvin Avenue, Leatherhead, Surrey, UK, 1982 Google Scholar[3] Avner Friedman, Variational principles and free-boundary problems, Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1982ix+710 84e:35153 0564.49002 Google Scholar[4] Avner Friedman and , Alessandro Torelli, A free boundary problem connected with non-steady filtration in porous media, Nonlinear Anal., 1 (1976/77), 503–545 10.1016/0362-546X(77)90015-3 58:29277 0373.76078 Avner Friedman and , Alessandro Torelli, Correction to the paper: “A free boundary problem connected with nonsteady filtration in porous media'', Nonlinear Anal., 2 (1978), 513–518 10.1016/0362-546X(78)90060-3 81m:76047 0429.76062 CrossrefGoogle Scholar[5] David Kinderlehrer and , Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, Vol. 88, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980xiv+313 81g:49013 0457.35001 Google Scholar[6] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969xx+554 41:4326 0189.40603 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Coupled Bulk-Surface Free Boundary Problems Arising from a Mathematical Model of Receptor-Ligand DynamicsCharles M. Elliott, Thomas Ranner, and Chandrasekhar VenkataramanSIAM Journal on Mathematical Analysis, Vol. 49, No. 1 | 8 February 2017AbstractPDF (6324 KB)On the State Estimation for Stochastic Distributed Parameter Systems with Dynamic Boundary ConditionsProceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications, Vol. 1998, No. 0 | 5 May 1998 Cross Ref A Class of Codimension-Two Free Boundary ProblemsS. D. Howison, J. D. Morgan, and J. R. OckendonSIAM Review, Vol. 39, No. 2 | 2 August 2006AbstractPDF (836 KB)Real-world free boundary problemsMathematics in Industrial Problems | 1 Jan 1991 Cross Ref A Class of Moving Boundary Problems Arising in IndustryApplied and Industrial Mathematics | 1 Jan 1991 Cross Ref A perturbation problem related to the highly compressible behaviour of a fluid in a thin porous layerApplicable Analysis, Vol. 33, No. 3-4 | 2 May 2007 Cross Ref Nonlinear semigroup approach to a class of evolution equations arising from percolation in sandbanksAnnali di Matematica Pura ed Applicata, Vol. 149, No. 1 | 1 Dec 1987 Cross Ref Error estimates for an approximation of a problem of percolation in gently sloping beachesCalcolo, Vol. 22, No. 3 | 1 Jul 1985 Cross Ref Volume 16, Issue 5| 1985SIAM Journal on Mathematical Analysis907-1119 History Submitted:24 April 1984Published online:17 February 2012 InformationCopyright © 1985 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0516071Article page range:pp. 941-954ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

Introduction Using finite-difference methods for European options is relatively straightforward, as there is no possibility of early exercise. As we have seen, the possibility of early exercise may lead to free boundaries. The chief problem with free boundaries, from the point of view of numerical analysis, is that we do not know where they are. This makes it difficult to impose the free boundary conditions, since we have to determine where to impose them as part of the solution procedure. (Recall that in Chapter 8 we simply imposed the boundary conditions at fixed grid points.) There are two distinct strategies for the numerical solution of free boundary problems. One is to attempt to track the free boundary as part of the time-stepping process. In the context of valuation of American options this is not a particularly attractive method, as the free boundary conditions are both implicit – that is, they do not give a direct expression for the free boundary or its time derivatives. We simply note the existence of such methods here, and refer the reader to the literature for a discussion of various boundary tracking strategies for implicit free boundary problems. The other strategy is to attempt to find a transformation that reduces the problem to a fixed boundary problem from which the free boundary can be inferred afterwards . There are many transformations that do this, but we consider only the particularly elegant method involving the use of the linear complementarity formulation.

This thesis is about shape optimization and optimal control for free boundary problems.We studied some free boundary problems, namely, a Bernoulli free boundary problem, a free boundary problem with surface tension, one with Ventcel type boundary conditions and another with Stokes free boundary conditions.We also performed the optimal control of the free boundary of this Bernoulli problem and the optimal control of the free boundary of the problem with surface tension.Theoretical results are presented for the free boundary problems, and the optimal control was performed for the Bernoulli free boundary problem and for the free boundary problem with free boundary conditions with surface tension.

The boundary element method applied to moving boundary problems

Boundary element method for a free third boundary problem modeling tumor growth with spectral accuracy

A moving mesh finite element method is studied for the numerical solution of Bernoulli free boundary problems. The method is based on the pseudo-transient continuation with which a moving boundary problem is constructed and its steady-state solution is taken as the solution of the underlying Bernoulli free boundary problem. The moving boundary problem is solved in a split manner at each time step: the moving boundary is updated with the Euler scheme, the interior mesh points are moved using a moving mesh method, and the corresponding initial-boundary value problem is solved using the linear finite element method. The method can take full advantages of both the pseudo-transient continuation and the moving mesh method. Particularly, it is able to move the mesh, free of tangling, to fit the varying domain for a variety of geometries no matter if they are convex or concave. Moreover, it is convergent towards steady state for a broad class of free boundary problems and initial guesses of the free boundary. Numerical examples for Bernoulli free boundary problems with constant and non-constant Bernoulli conditions and for nonlinear free boundary problems are presented to demonstrate the accuracy and robustness of the method and its ability to deal with various geometries and nonlinearities.

Diffusive KPP equations with free boundaries in time almost periodic environments: II. Spreading speeds and semi-wave solutions