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## Elliptic Equations (elliptic + equation)
## Selected Abstracts## ON AXISYMMETRIC TRAVELING WAVES AND RADIAL SOLUTIONS OF SEMI-LINEAR ELLIPTIC EQUATIONS NATURAL RESOURCE MODELING, Issue 3 2000THOMAS P. WITELSKIABSTRACT. Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementaryapproach to the detailed construction of axisymmetric diffusive interfaces in semi-linear elliptic equations. Solutions of the resulting non-autonomous radial differential equations can be expressed in terms of a slowlyvarying phase plane system. Special analytical results for the phase plane system are used to produce closed-form solutions for the asymptotic forms of the curved front solutions. These axisym-metric solutions are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensivelydiscuss a number of them, with a particular emphasis on connections to geometric models for the motion of interfaces. Related classical results for traveling waves in one-dimensional problems are also reviewed briefly. Manyof the results contained in this article are known, and in presenting known results, it is intended that this article be expositoryin nature, providing elementarydemonstrations of some of the central dynamical phenomena and mathematical techniques. It is hoped that the article serves as one possible avenue of entree to the literature on radiallysymmetric solutions of semilinear elliptic problems, especiallyto those articles in which more advanced mathematical theoryis developed. [source] ## A simplified v2,f model for near-wall turbulence INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2007M. M. RahmanAbstract A simplified version of the v2,f model is proposed that accounts for the distinct effects of low-Reynolds number and near-wall turbulence. It incorporates modified C,(1,2) coefficients to amplify the level of dissipation in non-equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. Unlike the conventional v2,f, it requires one additional equation (i.e. the elliptic equation for the elliptic relaxation parameter fµ) to be solved in conjunction with the k,, model. The scaling is evaluated from k in collaboration with an anisotropic coefficient Cv and fµ. Consequently, the model needs no boundary condition on and avoids free stream sensitivity. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Generation of Arbitrary Lagrangian,Eulerian (ALE) velocities, based on monitor functions, for the solution of compressible fluid equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005B. V. WellsAbstract A moving mesh method is outlined based on the use of monitor functions. The method is developed from a weak conservation principle. From this principle a conservation law for the mesh position is derived. Using the Helmholtz decomposition theorem, this conservation law can be converted into an elliptic equation for a mesh velocity potential. The moving mesh method is discretized using standard finite elements. Once the mesh velocities are obtained an arbitrary Lagrangian,Eulerian (ALE) (Journal of Computational Physics 1974; 14:227) fluid solver is used to update the solution on the adaptive mesh. Results are shown for the compressible Euler equations of gas dynamics in one and two spatial dimensions. Two monitor functions are used, the fluid density (which corresponds to a Lagrangian description), and a function which includes the density gradient. A variety of test problems are considered. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Chebyshev super spectral viscosity solution of a two-dimensional fluidized-bed model INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2003Scott A. SarraArticle first published online: 13 MAY 200Abstract The numerical solution of a model describing a two-dimensional fluidized bed by a Chebyshev super spectral viscosity (SSV) method is considered. The model is in the form of a hyperbolic system of conservation laws with a source term, coupled with an elliptic equation for determining a stream function. The coupled elliptic equation is solved by a finite-difference method. The mixed SSV/finite-difference method produces physically shaped bubbles, on a very coarse grid. Fine scale details, which were not present in previous finite-difference solutions, are present in the solution. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Homogenization of elliptic problems with the Dirichlet and Neumann conditions imposed on varying subsets MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2007Carmen Calvo-JuradoAbstract We study the asymptotic behaviour of the solution un of a linear elliptic equation posed in a fixed domain ,. The solution un is assumed to satisfy a Dirichlet boundary condition on ,n, where ,n is an arbitrary sequence of subsets of ,,, and a Neumman boundary condition on the remainder of ,,. We obtain a representation of the limit problem which is stable by homogenization and where it appears a generalized Fourier boundary condition. We also prove a corrector result. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Evolution completeness of separable solutions of non-linear diffusion equations in bounded domains MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2004V. A. GalaktionovAbstract As a basic example, we consider the porous medium equation (m > 1) (1) where , , ,N is a bounded domain with the smooth boundary ,,, and initial data . It is well-known from the 1970s that the PME admits separable solutions , where each ,k , 0 satisfies a non-linear elliptic equation . Existence of at least a countable subset , = {,k} of such non-linear eigenfunctions follows from the Lusternik,Schnirel'man variational theory from the 1930s. The first similarity pattern t,1/(m,1),0(x), where ,0 > 0 in ,, is known to be asymptotically stable as t , , and attracts all nontrivial solutions with u0 , 0 (Aronson and Peletier, 1981). We show that if , is discrete, then it is evolutionary complete, i.e. describes the asymptotics of arbitrary global solutions of the PME. For m = 1 (the heat equation), the evolution completeness follows from the completeness-closure of the orthonormal subset , = {,k} of eigenfunctions of the Laplacian , in L2. The analysis applies to the perturbed PME and to the p -Laplacian equations of second and higher order. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Note on a versatile Liapunov functional: applicability to an elliptic equation MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2002J. N. FlavinA novel, very effective Liapunov functional was used in previous papers to derive decay and asymptotic stability estimates (with respect to time) in a variety of thermal and thermo-mechanical contexts. The purpose of this note is to show that the versatility of this functional extends to certain non-linear elliptic boundary value problems in a right cylinder, the axial co-ordinate in this context replacing the time variable in the previous one. A steady-state temperature problem is considered with Dirichlet boundary conditions, the condition on the boundary being independent of the axial co-ordinate. The functional is used to obtain an estimate of the error committed in approximating the temperature field by the two-dimensional field induced by the boundary condition on the lateral surface. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Equilibrium problem for thermoelectroconductive body with the Signorini condition on the boundary MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2001D. HömbergAbstract We investigate a boundary value problem for a thermoelectroconductive body with the Signorini condition on the boundary, related to resistance welding. The mathematical model consists of an energy-balance equation coupled with an elliptic equation for the electric potential and a quasistatic momentum balance with a viscoelastic material law. We prove the existence of a weak solution to the model by using the Schauder fixed point theorem and classical results on pseudomonotone operators. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## On a semilinear elliptic equation with singular term and Hardy,Sobolev critical growth MATHEMATISCHE NACHRICHTEN, Issue 8 2007Jianqing ChenArticle first published online: 8 MAY 200Abstract In a previous work [6], we got an exact local behavior to the positive solutions of an elliptic equation. With the help of this exact local behavior, we obtain in this paper the existence of solutions of an equation with Hardy,Sobolev critical growth and singular term by using variational methods. The result obtained here, even in a particular case, relates with a partial (positive) answer to an open problem proposed in: A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177, 494,522 (2001). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Unified finite element discretizations of coupled Darcy,Stokes flow NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2009Trygve KarperAbstract In this article, we discuss some new finite element methods for flows which are governed by the linear stationary Stokes system on one part of the domain and by a second order elliptic equation derived from Darcy's law in the rest of the domain, and where the solutions in the two domains are coupled by proper interface conditions. All the methods proposed here utilize the same finite element spaces on the entire domain. In particular, we show how the coupled problem can be solved by using standard Stokes elements like the MINI element or the Taylor,Hood element in the entire domain. Furthermore, for all the methods the handling of the interface conditions are straightforward. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source] ## Anisotropic mesh adaptation for numerical solution of boundary value problems NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2004Vít DolejAbstract We present an efficient mesh adaptation algorithm that can be successfully applied to numerical solutions of a wide range of 2D problems of physics and engineering described by partial differential equations. We are interested in the numerical solution of a general boundary value problem discretized on triangular grids. We formulate a necessary condition for properties of the triangulation on which the discretization error is below the prescribed tolerance and control this necessary condition by the interpolation error. For a sufficiently smooth function, we recall the strategy how to construct the mesh on which the interpolation error is below the prescribed tolerance. Solving the boundary value problem we apply this strategy to the smoothed approximate solution. The novelty of the method lies in the smoothing procedure that, followed by the anisotropic mesh adaptation (AMA) algorithm, leads to the significant improvement of numerical results. We apply AMA to the numerical solution of an elliptic equation where the exact solution is known and demonstrate practical aspects of the adaptation procedure: how to control the ratio between the longest and the shortest edge of the triangulation and how to control the transition of the coarsest part of the mesh to the finest one if the two length scales of all the triangles are clearly different. An example of the use of AMA for the physically relevant numerical simulation of a geometrically challenging industrial problem (inviscid transonic flow around NACA0012 profile) is presented. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004. [source] ## Thermoelastic rolling contact problem with temperature dependent friction PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2008Andrzej ChudzikiewiczThe paper is concerned with the numerical solution of a thermoelastic rolling contact problem with wear. The friction between the bodies is governed by Coulomb law. A frictional heat generation and heat transfer across the contact surface as well as Archard's law of wear in contact zone are assumed. The friction coefficient is assumed to depend on temperature. In the paper quasistatic approach to solve this contact problem is employed. This approach is based on the assumption that for the observer moving with the rolling body the displacement of the supporting foundation is independent on time. The original thermoelastic contact problem described by the hyperbolic inequality governing the displacement and the parabolic equation governing the heat flow is transformed into elliptic inequality and elliptic equation, respectively. In order to solve numerically this system we decouple it into mechanical and thermal parts. Finite element method is used as a discretization method. Numerical examples showing the influence of the temperature dependent friction coefficient on the temperature distribution and the length of the contact zone are provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## A mass-conservative version of the semi-implicit semi-Lagrangian HIRLAM THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 635 2008P. H. LauritzenAbstract A mass-conservative version of the semi-implicit semi-Lagrangian High-Resolution Limited-Area Model (HIRLAM) is presented. The explicit continuity equation is solved with the so-called cell-integrated semi-Lagrangian (CISL) method. To allow for long time steps, the CISL scheme is coupled with a recently developed semi-implicit time-stepping scheme that involves the same non-complicated elliptic equation as in HIRLAM. Contrarily to the traditional semi-Lagrangian method, the trajectories are backward in the horizontal and forward in the vertical, i.e. cells moving with the flow depart from model layers and arrive in a regular column, and their vertical displacements are computed from continuity of mass and hydrostatic balance in the arrival column. This involves just two-dimensional upstream integrals and allows for a Lagrangian discretization of the energy conversion term in the thermodynamic equation. Preliminary validation of the new model version is performed using an idealized baroclinic wave test case. The accuracy of the new formulation of HIRLAM is comparable to the reference version though it is slightly more diffusive. A main finding is that the new discretization of the energy conversion term leads to more accurate simulations compared to the traditional ,Eulerian' treatment. Copyright © 2008 Royal Meteorological Society [source] ## A deterministic-control-based approach motion by curvature, COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 3 2006Robert KohnThe level-set formulation of motion by mean curvature is a degenerate parabolic equation. We show that its solution can be interpreted as the value function of a deterministic two-person game. More precisely, we give a family of discrete-time, two-person games whose value functions converge in the continuous-time limit to the solution of the motion-by-curvature PDE. For a convex domain, the boundary's "first arrival time" solves a degenerate elliptic equation; this corresponds, in our game-theoretic setting, to a minimum-exit-time problem. For a nonconvex domain the two-person game still makes sense; we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the "positive part of the curvature." These results are unexpected, because the value function of a deterministic control problem is normally the solution of a first-order Hamilton-Jacobi equation. Our situation is different because the usual first-order calculation is singular. © 2005 Wiley Periodicals, Inc. [source] ## Matched interface and boundary (MIB) method for the vibration analysis of plates INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 9 2009S. N. YuAbstract This paper proposes a novel approach, the matched interface and boundary (MIB) method, for the vibration analysis of rectangular plates with simply supported, clamped and free edges, and their arbitrary combinations. In previous work, the MIB method was developed for three-dimensional elliptic equations with arbitrarily complex material interfaces and geometric shapes. The present work generalizes the MIB method for eigenvalue problems in structural analysis with complex boundary conditions. The MIB method utilizes both uniform and non-uniform Cartesian grids. Fictitious values are utilized to facilitate the central finite difference schemes throughout the entire computational domain. Boundary conditions are enforced with fictitious values,a common practice used in the previous discrete singular convolution algorithm. An essential idea of the MIB method is to repeatedly use the boundary conditions to achieve arbitrarily high-order accuracy. A new feature in the proposed approach is the implementation of the cross derivatives in the free boundary conditions. The proposed method has a banded matrix. Nine different plates, particularly those with free edges and free corners, are employed to validate the proposed method. The performance of the proposed method is compared with that of other established methods. Convergence and comparison studies indicate that the proposed MIB method works very well for the vibration analysis of plates. In particular, modal bending moments and shear forces predicted by the proposed method vanish at boundaries for free edges. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Large eddy simulation of turbulent flows via domain decomposition techniques. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2005Part 1: theoryAbstract The present paper discusses large eddy simulations of incompressible turbulent flows in complex geometries. Attention is focused on the application of the Schur complement method for the solution of the elliptic equations arising from the fractional step procedure and/or the semi-implicit discretization of the momentum equations in velocity,pressure representation. Fast direct and iterative Poisson solvers are compared and their global efficiency evaluated both in serial and parallel architecture environments for model problems of physical relevance. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Finite energy solutions of self-adjoint elliptic mixed boundary value problems MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2010Giles AuchmutyAbstract This paper describes existence, uniqueness and special eigenfunction representations of H1 -solutions of second order, self-adjoint, elliptic equations with both interior and boundary source terms. The equations are posed on bounded regions with Dirichlet conditions on part of the boundary and Neumann conditions on the complement. The system is decomposed into separate problems defined on orthogonal subspaces of H1(,). One problem involves the equation with the interior source term and the Neumann data. The other problem just involves the homogeneous equation with Dirichlet data. Spectral representations of the solution operators for each of these problems are found. The solutions are described using bases that are, respectively, eigenfunctions of the differential operator with mixed null boundary conditions, and certain mixed Steklov eigenfunctions. These series converge strongly in H1(,). Necessary and sufficient conditions for the Dirichlet part of the boundary data to have a finite energy extension are described. The solutions for a problem that models a cylindrical capacitor is found explicitly. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains MATHEMATISCHE NACHRICHTEN, Issue 12 2008Annegret GlitzkyAbstract We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain ,0 of the domain of definition , of the energy balance equation and of the Poisson equation. Here ,0 corresponds to the region of semiconducting material, , \ ,0 represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a W1,p -regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On a semilinear elliptic equation with singular term and Hardy,Sobolev critical growth MATHEMATISCHE NACHRICHTEN, Issue 8 2007Jianqing ChenArticle first published online: 8 MAY 200Abstract In a previous work [6], we got an exact local behavior to the positive solutions of an elliptic equation. With the help of this exact local behavior, we obtain in this paper the existence of solutions of an equation with Hardy,Sobolev critical growth and singular term by using variational methods. The result obtained here, even in a particular case, relates with a partial (positive) answer to an open problem proposed in: A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177, 494,522 (2001). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Best Sobolev constants and quasi-linear elliptic equations with critical growth on spheres MATHEMATISCHE NACHRICHTEN, Issue 12-13 2005C. BandleAbstract Sharp existence and nonexistence results for positive solutions of quasilinear elliptic equations with critical growth in geodesic balls on spheres are established. The arguments are based on Pohozaev type identities and asymptotic estimates for Emden,Fowler type equations. By means of spherical symmetrization and the concentration-compactness principle existence and nonexistence results for general domains on spheres are obtained. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## An iterative procedure for solving a Cauchy problem for second order elliptic equations MATHEMATISCHE NACHRICHTEN, Issue 1 2004Tomas JohanssonArticle first published online: 14 JUL 200Abstract An iterative method for reconstruction of solutions to second order elliptic equations by Cauchy data given on a part of the boundary, is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the elliptic operator and its adjoint. The convergence proof of this method in a weighted L2 space is included. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Comparison results for nonlinear elliptic equations with lower,order terms MATHEMATISCHE NACHRICHTEN, Issue 1 2003Vincenzo FeroneAbstract We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) + g(x, u) = f, where the principal term is a Leray,Lions operator defined on and g(x, u) is a term having the same sign as u and satisfying suitable growth assumptions. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric. [source] ## ON AXISYMMETRIC TRAVELING WAVES AND RADIAL SOLUTIONS OF SEMI-LINEAR ELLIPTIC EQUATIONS NATURAL RESOURCE MODELING, Issue 3 2000THOMAS P. WITELSKIABSTRACT. Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementaryapproach to the detailed construction of axisymmetric diffusive interfaces in semi-linear elliptic equations. Solutions of the resulting non-autonomous radial differential equations can be expressed in terms of a slowlyvarying phase plane system. Special analytical results for the phase plane system are used to produce closed-form solutions for the asymptotic forms of the curved front solutions. These axisym-metric solutions are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensivelydiscuss a number of them, with a particular emphasis on connections to geometric models for the motion of interfaces. Related classical results for traveling waves in one-dimensional problems are also reviewed briefly. Manyof the results contained in this article are known, and in presenting known results, it is intended that this article be expositoryin nature, providing elementarydemonstrations of some of the central dynamical phenomena and mathematical techniques. It is hoped that the article serves as one possible avenue of entree to the literature on radiallysymmetric solutions of semilinear elliptic problems, especiallyto those articles in which more advanced mathematical theoryis developed. [source] ## A robust multilevel approach for minimizing H(div)-dominated functionals in an H1 -conforming finite element space NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2004Travis M. AustinAbstract The standard multigrid algorithm is widely known to yield optimal convergence whenever all high-frequency error components correspond to large relative eigenvalues. This property guarantees that smoothers like Gauss,Seidel and Jacobi will significantly dampen all the high-frequency error components, and thus, produce a smooth error. This has been established for matrices generated from standard discretizations of most elliptic equations. In this paper, we address a system of equations that is generated from a perturbation of the non-elliptic operator I-grad div by a negative , ,. For ,near to one, this operator is elliptic, but as ,approaches zero, the operator becomes non-elliptic as it is dominated by its non-elliptic part. Previous research on the non-elliptic part has revealed that discretizing I-grad div with the proper finite element space allows one to define a robust geometric multigrid algorithm. The robustness of the multigrid algorithm depends on a relaxation operator that yields a smooth error. We use this research to assist in developing a robust discretization and solution method for the perturbed problem. To this end, we introduce a new finite element space for tensor product meshes that is used in the discretization, and a relaxation operator that succeeds in dampening all high-frequency error components. The success of the corresponding multigrid algorithm is first demonstrated by numerical results that quantitatively imply convergence for any ,is bounded by the convergence for ,equal to zero. Then we prove that convergence of this multigrid algorithm for the case of , equal to zero is independent of mesh size. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## On the mixed finite element method with Lagrange multipliers NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2003Ivo BabuAbstract In this note we analyze a modified mixed finite element method for second-order elliptic equations in divergence form. As a model we consider the Poisson problem with mixed boundary conditions in a polygonal domain of R2. The Neumann (essential) condition is imposed here in a weak sense, which yields the introduction of a Lagrange multiplier given by the trace of the solution on the corresponding boundary. This approach allows to handle nonhomogeneous Neumann boundary conditions, theoretically and computationally, in an alternative and usually easier way. Then we utilize the classical Babu,ka-Brezzi theory to show that the resulting mixed variational formulation is well posed. In addition, we use Raviart-Thomas spaces to define the associated finite element method and, applying some elliptic regularity results, we prove the stability, unique solvability, and convergence of this discrete scheme, under appropriate assumptions on the mesh sizes. Finally, we provide numerical results illustrating the performance of the algorithm for smooth and singular problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 192,210, 2003 [source] ## Dirichlet duality and the nonlinear Dirichlet problem COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 3 2009F. Reese HarveyWe study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F(Hess u) = 0 on a smoothly bounded domain , , ,n. In our approach the equation is replaced by a subset F , Sym2(,n) of the symmetric n × n matrices with ,F , {F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric "F -convexity" assumption on the boundary ,,. We also study the topological structure of F -convex domains and prove a theorem of Andreotti-Frankel type. Two key ingredients in the analysis are the use of "subaffine functions" and "Dirichlet duality." Associated to F is a Dirichlet dual set F, that gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F, is F, and in the analysis the roles of F and F, are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: all branches of the homogeneous Monge-Ampère equation over ,, ,, and ,; equations appearing naturally in calibrated geometry, Lagrangian geometry, and p -convex Riemannian geometry; and all branches of the special Lagrangian potential equation. © 2008 Wiley Periodicals, Inc. [source] ## A constant rank theorem for solutions of fully nonlinear elliptic equations COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 12 2007Luis A. CaffarelliFirst page of article [source] ## Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs, COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2007Eitan TadmorWe prove in this paper new velocity-averaging results for second-order multidimensional equations of the general form ,(,x, v)f(x, v) = g(x, v) where ,(,x, v) := a(v) · ,x , , · b(v),x. These results quantify the Sobolev regularity of the averages, ,vf(x, v),(v)dv, in terms of the nondegeneracy of the set {v: |,(i,, v)| , ,} and the mere integrability of the data, (f, g) , (L, L). Velocity averaging is then used to study the regularizing effect in quasi-linear second-order equations, ,(,x, ,), = S(,), which use their underlying kinetic formulations, ,(,x, v),, = gS. In particular, we improve previous regularity statements for nonlinear conservation laws, and we derive completely new regularity results for convection-diffusion and elliptic equations driven by degenerate, nonisotropic diffusion. © 2007 Wiley Periodicals, Inc. [source] |