Interior Regularity Research Articles

The classical plateau problem and the topology of three-dimensional manifolds: The embedding of the solution given by Douglas-Morrey and an analytic proof of Dehn's Lemma

LET y be a rectifiable Jordan curve in three-dimensional euclidean space. Answering an old question, whether y can bound a surface with minimal area, Douglas [l I] and Rad6[45] (independently) found a minimal surface spanning y which is parametrized by the disk. This minimal surface has minimal area among all Lipschitz maps from the disk into R3 which span y. The question whether this solution has branch points or not was finally settled by Osserman[42], who proved that there are no interior “true” branch points, and by Gulliver[lS], who proved that there are no interior “false” branch points. In 1948, Morrey[35] devised a new method to solve the Plateau problem for a map from the disk into a “homogeneously regular*’ Riemannian manifold. Moreover, he proved the interior regularity of the map in case the ambient manifold is regular, and that the map is real analytic if the ambient manifold is real analytic. The arguments of Osserman and Gulliver in addition show that Morrey’s solution has no interior branch point when the ambient manifold is three-dimensional. In 1951, Lewy[29] showed that if the Jordan curve y is also real analytic, in a real analytic manifold, then any minimal surface with boundary y is real analytic up to the boundary. Hence in this case Morrey’s map is real analytic map on the closed disk. (For a proof of Lewy’s Theorem in a general real analytic manifold, see HiIdebrandt[23].) In 1969, Hildebrandt[23] proved that the Douglas solution is smooth up to the boundary if the Jordan curve is smooth and regular. (Further improvements are due to Kinderlehrer [25], Nitsche [40], and Warschawski[55].) In [22], Heinz and Hildebrandt extended Hildebrandt’s result to minimal surfaces in general Riemannian manifolds. Once we have boundary regularity, it makes sense to ask whether Douglas’ or Morrey’s solution of the Plateau problem has a boundary branch point or not, To date, this problem has not been settled. The first partial result in this direction is due to Nitsche[40], who showed that there are only a finite number of boundary branch points for minimal surfaces with smooth boundary in R3. This was then generalized by Heinz and Hildebrandt to smooth manifolds. Gulliver and Lesley[l6] have also observed that the Douglas-Morrey solution for a real analytic curve in a real analytic manifold has no boundary branch point, using the previously mentioned result of Lewy. Despite all these results, an interesting topological question remained unsolved, namely, under what conditions is the Douglas solution an embedded surface? It has been generally conjectured that when the Jordan curve is extremal, i.e. lies on the

Interior regularity operators

ПустьР - линейный диф ференциальный опера тор с достаточно гладкими коэффициентами. По определению,P явля ется оператором внут ренней регулярности на ω ⊂R n т огда и только тогда, когда\(u \in B_{p,k_{ - N} }^{loc} (\Omega )\) и ω′⊂ω из условия\(Pu \in B_{p,k_s }^{loc} (\Omega ')\) вытекает, что\(u \in B_{p,k_s k}^{loc} (\Omega ')\), где −N+1≦s≦N. Соотве тствующий пример: $$Pu = - \Delta u + u c k(\xi ) = \xi _1^2 + \ldots + \xi _n^2 + 1.$$ Указанные операторы характеризуются в ра боте в терминах априорных н еравенств. До− казывается также сущ ествование локальны х фундаментальных реш ений для оператора, со пряженного кP, а также его гладкос ть вне диагонали. Эти результаты являются аналогами соответствующих рез ультатов для гипоэлл иптических операторов.

Stochastic representation and interior regularity of solutions of degenerate elliptic-parabolic equations

Stochastic representation and interior regularity of solutions of degenerate elliptic-parabolic equations

Zur Regularität der quasistatischen elasto-viskoplastischen Verschiebungen und Spannungen

We prove some interior regularity properties of the displacements and stresses for a quasistatic elasto-visco-plastic modell in continuum mechanics.

Interior regularity of solutions to systems of variational inequalities

Interior regularity of solutions to systems of variational inequalities

The Equivariant Plateau Problem and Interior Regularity

The Equivariant Plateau Problem and Interior Regularity

The equivariant Plateau problem and interior regularity

Let M ⊂ R n M \subset {{\text {R}}^n} be a compact submanifold of Euclidean space which is invariant by a compact group G ⊂ S O ( n ) G \subset SO(n) . When dim ⁡ ( M ) = n − 2 \dim (M) = n - 2 , it is shown that there always exists a solution to the Plateau problem for M M which is invariant by G G and, furthermore, that uniqueness of this solution among G G -invariant currents implies uniqueness in general. This result motivates the subsequent study of the Plateau problem for M M within the class of G G -invariant integral currents. It is shown that this equivariant problem reduces to the study of a corresponding Plateau problem in the orbit space R / G {\text {R}}/G where, for “big” groups, questions of uniqueness and regularity are simplified. The method is then applied to prove that for a constellation of explicit manifolds M M , the cone C ( M ) = { t x ; x ∈ M C(M) = \{ tx;x \in M and 0 ⩽ t ⩽ 1 } 0 \leqslant t \leqslant 1\} is the unique solution to the Plateau problem for M M , (Thus, there is no hope for general interior regularity of solutions in codimension one.) These manifolds include the original examples of type S n × S n ⊂ R 2 n + 2 , n ⩾ 3 {S^n} \times {S^n} \subset {{\text {R}}^{2n + 2}},n \geqslant 3 , due to Bombieri, DeGiorgi, Giusti and Simons. They also include a new example in R 8 {{\text {R}}^8} and examples in R n {{\text {R}}^n} for n ⩾ 10 n \geqslant 10 with any prescribed Betti number nonzero.

Pointwise Bounds for Discrete Green’s Functions

Previous article Next article Pointwise Bounds for Discrete Green’s FunctionsJ. H. Bramble and V. ThoméeJ. H. Bramble and V. Thoméehttps://doi.org/10.1137/0706053PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. H. Bramble, On the convergence of difference approximations to weak solutions of Dirichlet's problem, Numer. Math., 13 (1969), 101–111 10.1007/BF02163229 MR0250495 0196.17803 CrossrefISIGoogle Scholar[2] James H. Bramble, On the convergence of difference approximations for second order uniformly elliptic operatorsNumerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968), SIAM-AMS Proc., Vol. II, Amer. Math. Soc., Providence, R. I., 1970, 201–209 MR0260200 0234.65086 Google Scholar[3] J. H. Bramble, , B. E. Hubbard and , Vidar Thomée, Convergence estimates for essentially positive type discrete Dirichlet problems, Math. Comp., 23 (1969), 695–709 MR0266444 0217.21902 CrossrefISIGoogle Scholar[4] J. H. Bramble, , B. E. Hubbard and , M. Zlámal, Discrete analogues of the Dirichlet problem with isolated singularities, SIAM J. Numer. Anal., 5 (1968), 1–25 10.1137/0705001 MR0239770 0176.46902 LinkISIGoogle Scholar[5] James R. Kuttler, Finite difference approximations for eigenvalues of uniformly elliptic operators, SIAM J. Numer. Anal., 7 (1970), 206–232 10.1137/0707014 MR0273841 0208.42601 LinkISIGoogle Scholar[6] Pentti Laasonen, On the solution of Poisson's difference equation, J. Assoc. Comput. Mach., 5 (1958), 370–382 MR0121999 0087.12202 CrossrefISIGoogle Scholar[7] Vidar Thomée and , Bertil Westergren, Elliptic difference equations and interior regularity, Numer. Math., 11 (1968), 196–210 MR0224303 0159.38204 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Asymptotics of the heat kernels on 2D latticesAsymptotic Analysis, Vol. 112, No. 1-2 | 6 Mar 2019 Cross Ref On Solving the Singular System Arisen from Poisson Equation with Neumann Boundary ConditionJournal of Scientific Computing, Vol. 69, No. 1 | 4 April 2016 Cross Ref Petrov–Galerkin finite element approximation for the three dimensional well modelJournal of Computational and Applied Mathematics, Vol. 277 | 1 Mar 2015 Cross Ref A 3-dimensional well model in the flow transport through porous mediaApplied Mathematical Modelling, Vol. 38, No. 21-22 | 1 Nov 2014 Cross Ref On the decay of the inverse of matrices that are sum of Kronecker productsLinear Algebra and its Applications, Vol. 452 | 1 Jul 2014 Cross Ref On Sojourn Times in the M/ M/1-PS Model, Conditioned on the Number of Other UsersApplied Mathematics Research eXpress, Vol. 11 | 19 February 2010 Cross Ref A kernel-free boundary integral method for elliptic boundary value problemsJournal of Computational Physics, Vol. 227, No. 2 | 1 Dec 2007 Cross Ref On the accuracy of finite difference methods for elliptic problems with interfacesCommunications in Applied Mathematics and Computational Science, Vol. 1, No. 1 | 31 December 2006 Cross Ref Least squares methods for elliptic systemsMathematics of Computation, Vol. 44, No. 169 | 1 January 1985 Cross Ref L�sung des Dirichletproblems und Konvergenz der Differenzenapproximation f�r elliptische Differentialgleichungen zweiter OrdnungMathematische Zeitschrift, Vol. 131, No. 3 | 1 Sep 1973 Cross Ref Discrete Green’s functionsMathematics of Computation, Vol. 27, No. 121 | 1 January 1973 Cross Ref The distribution of the eigenvalues of the discrete LaplacianBIT, Vol. 12, No. 4 | 1 Dec 1972 Cross Ref A fourth-order finite-difference approximation for the fixed membrane eigenproblemMathematics of Computation, Vol. 25, No. 114 | 1 January 1971 Cross Ref Volume 6, Issue 4| 1969SIAM Journal on Numerical Analysis523-616 History Submitted:14 November 1968Published online:14 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0706053Article page range:pp. 583-590ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

Remarks on Almgren’s interior regularity theorems

Remarks on Almgren’s interior regularity theorems

Elliptic difference equations and interior regularity

Letu be a solution of an elliptic differential equation and letu h be a solution of a corresponding elliptic difference equation. It is proved that ifu h converges tou at a certain rate as the mesh-widthh tends to zero, then the appropriate difference quotient ofu h converges to the corresponding derivative ofu at the same rate whenh tends to zero.

Some Interior Regularity Theorems for Minimal Surfaces and an Extension of Bernstein's Theorem

Some Interior Regularity Theorems for Minimal Surfaces and an Extension of Bernstein's Theorem

On the interior regularity of weak solutions of the Navier-Stokes equations

On the interior regularity of weak solutions of the Navier-Stokes equations

Interior regularity of weak solutions of the time-dependent Navier-Stokes equation

Interior regularity of weak solutions of the time-dependent Navier-Stokes equation

On interior regularity of the solutions of partial differential equations

On interior regularity of the solutions of partial differential equations

Partial Differential Equations and Monge-Kantorovich Mass Transfer

2. Heuristics 2.1 Geometry of optimal transport 2.2 Lagrange multipliers 3. Optimal mass transport, polar factorization 3.1 Solution of dual problem 3.2 Existence of optimal mass transfer plan 3.3 Polar factorization of vector fields 4. Regularity 4.1 Solving the Monge–Ampere equation 4.2 Examples 4.3 Interior regularity for convex targets 4.4 Boundary regularity for convex domain and target 5. Application: Nonlinear interpolation 6. Application: Time-step minimization and nonlinear diffusion 6.1 Discrete time approximation 6.2 Euler–Lagrange equation