Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted Lq-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt $A_{\infty }$ A ∞ -class. In the Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the Ap-range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.

[1]  B. Schulze,et al.  Index theory of elliptic boundary problems , 1982 .

[2]  Bui Huy Qui.  Weighted Besov and Triebel spaces: interpolation by the real method , 1982 .

[3]  G. Grubb.  Distributions and Operators , 2008 .

[4]  M. Meyries,et al.  Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions , 2012, 1202.3875.

[5]  Yoshikazu Giga,et al.  Analyticity of the semigroup generated by the Stokes operator inLr spaces , 1981 .

[6]  N. Lindemulder.  Second Order Operators Subject to Dirichlet Boundary Conditions in Weighted Triebel-Lizorkin Spaces: Parabolic Problems. , 2018, 1812.05462.

[7]  G. Dore.  $H^{∞}$ functional calculus in real interpolation spaces, II , 1999 .

[8]  D. Haroske,et al.  Entropy numbers of embeddings of function spaces with Muckenhoupt weights, III. Some limiting cases , 2011 .

[9]  W. Sickel,et al.  Complex interpolation of weighted Besov and Lizorkin-Triebel spaces , 2012, 1212.1614.

[10]  G. Dore.  Maximal regularity in $L^p$ spaces for an abstract Cauchy problem , 2000, Advances in Differential Equations.

[11]  L. B. D. Monvel.  Boundary problems for pseudo-differential operators , 1971 .

[12]  R. Farwig,et al.  Weighted $L^{q}$-theory for the Stokes resolvent in exterior domains , 1997 .

[13]  N. Lindemulder,et al.  An intersection representation for a class of anisotropic vector-valued function spaces , 2019, J. Approx. Theory.

[14]  Jan Pruess,et al.  On Critical Spaces for the Navier–Stokes Equations , 2017, 1703.08714.

[15]  N. Lindemulder.  Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions , 2017, Journal of Evolution Equations.

[16]  Hongjie Dong,et al.  Higher-order elliptic and parabolic equations with VMO assumptions and general boundary conditions , 2017, 1702.03254.

[17]  M. Veraar,et al.  Complex interpolation with Dirichlet boundary conditions on the half line , 2017, Mathematische Nachrichten.

[18]  Jan Prüss,et al.  Moving Interfaces and Quasilinear Parabolic Evolution Equations , 2016 .

[19]  G. Dore,et al.  H ∞ functional calculus in real interpolation spaces , 2022 .

[20]  Hongjie Dong,et al.  Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces , 2014, 1403.2459.

[21]  N. Krylov.  Weighted sobolev spaces and laplace's equation and the heat equations in a half space , 1999 .

[22]  Y. Roitberg.  Elliptic Boundary Value Problems in the Spaces of Distributions , 1996 .

[23]  W. Arendt,et al.  Maximal Lp-regularity for parabolic and elliptic equations on the line , 2006 .

[24]  D. Haroske,et al.  Entropy and Approximation Numbers of Embeddings of Function Spaces with Muckenhoupt Weights, I , 2008 .

[25]  B. Scharf.  Atomic representations in function spaces and applications to pointwise multipliers and diffeomorphisms, a new approach , 2011, 1111.6812.

[26]  R. Chill,et al.  Singular integral operators with operator-valued kernels, and extrapolation of maximal regularity into rearrangement invariant Banach function spaces , 2014 .

[27]  Singular Green operators and their spectral asymptotics , 1984 .

[28]  J. Johnsen.  Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel--Lizorkin spaces , 1996, 1704.08555.

[29]  J. Prüss,et al.  An Operator-Valued Transference Principle and Maximal Regularity on Vector-Valued Lp -spaces , 2019, Evolution Equations and Their Applications in Physical and Life Sciences.

[30]  Comportement d'un opérateur pseudo-différentiel sur une variété à bord: II. Pseudo-noyaux de Poisson , 1966 .

[31]  M. Veraar,et al.  Sharp embedding results for spaces of smooth functions with power weights , 2011, 1112.5388.

[32]  L. Weis,et al.  Maximal Lp-regularity for Parabolic Equations, Fourier Multiplier Theorems and $H^\infty$-functional Calculus , 2004 .

[33]  J. Wloka,et al.  Partial differential equations , 1987 .

[34]  G. Simonett,et al.  Critical spaces for quasilinear parabolic evolution equations and applications , 2017, 1708.08550.

[35]  G. Grubb,et al.  A global calculus of parameter-dependent pseudodifferential boundary problems inLp Sobolev spaces , 1993 .

[36]  Lutz Weis,et al.  Operator–valued Fourier multiplier theorems and maximal $L_p$-regularity , 2001 .

[37]  J. Prüss,et al.  On quasilinear parabolic evolution equations in weighted Lp-spaces , 2009 .

[38]  M. Veraar,et al.  Characterization of a class of embeddings for function spaces with Muckenhoupt weights , 2014, 1409.2396.

[39]  Giovanni Dore,et al.  On the closedness of the sum of two closed operators , 1987 .

[40]  G. Simonett,et al.  Maximal regularity for evolution equations in weighted Lp-spaces , 2004 .

[41]  Jan Pruess,et al.  On quasilinear parabolic evolution equations in weighted Lp-spaces II , 2009, 0909.1480.

[42]  M. H. Taibleson,et al.  A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces , 1996 .

[43]  N. Kalton,et al.  The H ∞ −calculus and sums of closed operators , 2001 .

[44]  N. Kalton,et al.  Perturbation and Interpolation Theorems for the H∞-Calculus with Applications to Differential Operators , 2006 .

[45]  Kyeong-Hun Kim.  Lq(Lp)-Theory of parabolic PDEs with variable coefficients , 2008 .

[46]  J. Wloka,et al.  Partial differential equations: Strongly elliptic differential operators and the method of variations , 1987 .

[47]  R. Denk,et al.  General Parabolic Mixed Order Systems in Lp and Applications , 2013 .

[48]  Alexander Mielke.  Über maximaleLp-Regularität für Differentialgleichungen in Banach- und Hilbert-Räumen , 1987 .

[49]  R. Sowers.  Multidimensional Reaction-Diffusion Equations with White Noise Boundary Perturbations , 1994 .

[50]  M. Veraar,et al.  Pointwise Multiplication on Vector-Valued Function Spaces with Power Weights , 2013, 1311.7404.

[51]  D. Haroske,et al.  Atomic decompositions of function spaces with Muckenhoupt weights, and some relation to fractal analysis , 2008 .

[52]  M. Meyries.  Maximal Regularity in Weighted Spaces, Nonlinear Boundary Conditions, and Global Attractors , 2010 .

[53]  R. Chill,et al.  Real interpolation with weighted rearrangement invariant Banach function spaces , 2015, 1503.05718.

[54]  Vyacheslav S. Rychkov,et al.  On Restrictions and Extensions of the Besov and Triebel–Lizorkin Spaces with Respect to Lipschitz Domains , 1999 .

[55]  Hongjie Dong,et al.  Higher-order parabolic equations with VMO assumptions and general boundary conditions with variable leading coefficients , 2017, 1709.04337.

[56]  The $H^{\infty}-$calculus and sums of closed operators , 2000, math/0010155.

[57]  G. Pólya,et al.  Functions of One Complex Variable , 1998 .

[58]  G. Grubb.  Pseudo-differential boundary problems in Lp, spaces , 1990 .

[59]  E. Alòs,et al.  STABILITY FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH DIRICHLET WHITE-NOISE BOUNDARY CONDITIONS , 2002 .

[60]  Petru A. Cioica-Licht,et al.  On the regularity of the stochastic heat equation on polygonal domains in R2 , 2018, Journal of Differential Equations.

[61]  L. B. Monvel.  Comportement d'un opérateur pseudo-différentiel sur une variété à bord , 1966 .

[62]  J. Prüss,et al.  Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data , 2007 .

[63]  W. Sickel,et al.  On the trace problem for Lizorkin–Triebel spaces with mixed norms , 2008, 1702.00712.

[64]  Winfried Sickel,et al.  Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations , 1996, de Gruyter series in nonlinear analysis and applications.

[65]  T. Hytönen,et al.  Weighted estimates for operator-valued Fourier multipliers , 2018, Collectanea Mathematica.

[66]  L. Weis,et al.  Erratum to: Perturbation and interpolation theorems for the H∞-calculus with applications to differential operators , 2013, Mathematische Annalen.

[67]  M. Veraar,et al.  The heat equation with rough boundary conditions and holomorphic functional calculus , 2018, Journal of Differential Equations.

[68]  E. Schrohe.  A Short Introduction to Boutet de Monvel’s Calculus , 2001 .

[69]  R. T. Seeley,et al.  Extension of ^{∞} functions defined in a half space , 1964 .

[70]  Herbert Amann,et al.  Linear and Quasilinear Parabolic Problems , 2019, Monographs in Mathematics.

[71]  M. Veraar,et al.  Traces and embeddings of anisotropic function spaces , 2012, 1207.1044.

[72]  G. Simonett,et al.  Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations , 2001 .

[73]  V. Maz'ya,et al.  Higher regularity in the classical layer potential theory for Lipschitz domains , 2005 .

[74]  Felix Hummel.  Boundary value problems of elliptic and parabolic type with boundary data of negative regularity , 2021, Journal of Evolution Equations.

[75]  Robert Denk,et al.  Fourier multipliers and problems of elliptic and parabolic type , 2003 .

[76]  R. Denk,et al.  Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity , 2016 .

[77]  G. FABBRI,et al.  An LQ Problem for the Heat Equation on the Halfline with Dirichlet Boundary Control and Noise , 2008, SIAM J. Control. Optim..

[78]  Nicolai V. Krylov,et al.  The Heat Equation in Lq((0, T), Lp)-Spaces with Weights , 2001, SIAM J. Math. Anal..

[79]  D. Haroske,et al.  ENTROPY AND APPROXIMATION NUMBERS OF EMBEDDINGS OF FUNCTION SPACES WITH MUCKENHOUPT WEIGHTS, II. GENERAL WEIGHTS , 2011 .

[80]  Z. Brzeźniak,et al.  Second order PDEs with Dirichlet white noise boundary conditions , 2013, 1305.5324.

[81]  Loukas Grafakos,et al.  Modern Fourier Analysis , 2008 .