Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 7 (2014), 1649-1682.

Local and nonlocal boundary conditions for $\mu$-transmission and fractional elliptic pseudodifferential operators

Gerd Grubb

Full-text: Open access

Abstract

A classical pseudodifferential operator P on n satisfies the μ-transmission condition relative to a smooth open subset Ω when the symbol terms have a certain twisted parity on the normal to Ω. As shown recently by the author, this condition assures solvability of Dirichlet-type boundary problems for P in full scales of Sobolev spaces with a singularity dμk, d(x)= dist(x,Ω). Examples include fractional Laplacians (Δ)a and complex powers of strongly elliptic PDE.

We now introduce new boundary conditions, of Neumann type, or, more generally, nonlocal type. It is also shown how problems with data on nΩ reduce to problems supported on Ω¯, and how the so-called “large” solutions arise. Moreover, the results are extended to general function spaces Fp,qs and Bp,qs, including Hölder–Zygmund spaces B,s. This leads to optimal Hölder estimates, e.g., for Dirichlet solutions of (Δ)au=fL(Ω), udaCa(Ω¯) when 0<a<1, a12.

Article information

Source
Anal. PDE, Volume 7, Number 7 (2014), 1649-1682.

Dates
Received: 8 April 2014
Revised: 25 August 2014
Accepted: 23 September 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731607

Digital Object Identifier
doi:10.2140/apde.2014.7.1649

Mathematical Reviews number (MathSciNet)
MR3293447

Zentralblatt MATH identifier
1317.35310

Subjects
Primary: 35S15: Boundary value problems for pseudodifferential operators
Secondary: 45E99: None of the above, but in this section 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx]

Keywords
fractional Laplacian boundary regularity Dirichlet and Neumann conditions large solutions Hölder–Zygmund spaces Besov–Triebel–Lizorkin spaces transmission properties elliptic pseudodifferential operators singular integral operators

Citation

Grubb, Gerd. Local and nonlocal boundary conditions for $\mu$-transmission and fractional elliptic pseudodifferential operators. Anal. PDE 7 (2014), no. 7, 1649--1682. doi:10.2140/apde.2014.7.1649. https://projecteuclid.org/euclid.apde/1513731607


Export citation

References

  • N. Abatangelo, “Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian”, preprint, 2013.
  • H. Abels, “Pseudodifferential boundary value problems with non-smooth coefficients”, Comm. Partial Differential Equations 30:10-12 (2005), 1463–1503.
  • P. Albin and R. Melrose, “Fredholm realizations of elliptic symbols on manifolds with boundary”, J. Reine Angew. Math. 627 (2009), 155–181.
  • L. Caffarelli and L. Silvestre, “Regularity theory for fully nonlinear integro-differential equations”, Comm. Pure Appl. Math. 62:5 (2009), 597–638.
  • O. Chkadua and R. Duduchava, “Pseudodifferential equations on manifolds with boundary: Fredholm property and asymptotic”, Math. Nachr. 222 (2001), 79–139.
  • G. I. Eskin, Boundary value problems for elliptic pseudodifferential equations, Translations of Mathematical Monographs 52, American Mathematical Society, Providence, R.I., 1981. Translated from the Russian by S. Smith.
  • M. Felsinger, M. Kassmann, and P. Voigt, “The Dirichlet problem for nonlocal operators”, Math. Zeitschrift (online publication November 2014).
  • R. L. Frank and L. Geisinger, “Refined semiclassical asymptotics for fractional powers of the Laplace operator”, J. Reine Angew. Math. (online publication January 2014).
  • G. Grubb, “Singular Green operators and their spectral asymptotics”, Duke Math. J. 51:3 (1984), 477–528.
  • G. Grubb, “Pseudo-differential boundary problems in $L\sb p$ spaces”, Comm. Partial Differential Equations 15:3 (1990), 289–340.
  • G. Grubb, Functional calculus of pseudodifferential boundary problems, 2nd ed., Progress in Mathematics 65, Birkhäuser, Boston, MA, 1996.
  • G. Grubb, Distributions and operators, Graduate Texts in Mathematics 252, Springer, New York, 2009.
  • G. Grubb, “Spectral asymptotics for nonsmooth singular Green operators”, Comm. Partial Differential Equations 39:3 (2014), 530–573. With an appendix by H. Abels.
  • G. Grubb, “Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators”, Adv. Math. 268 (2015), 478–528.
  • G. Grubb, “Spectral results for mixed problems and fractional elliptic operators”, J. Math. Anal. Appl. 421:2 (2015), 1616–1634.
  • G. Grubb and L. H örmander, “The transmission property”, Math. Scand. 67:2 (1990), 273–289.
  • G. Harutyunyan and B.-W. Schulze, Elliptic mixed, transmission and singular crack problems, EMS Tracts in Mathematics 4, European Mathematical Society (EMS), Zürich, 2008.
  • W. Hoh and N. Jacob, “On the Dirichlet problem for pseudodifferential operators generating Feller semigroups”, J. Funct. Anal. 137:1 (1996), 19–48.
  • L. Hörmander, “Chapter II: Boundary problems for “classical” pseudo-differential operators”, unpublished lecture notes at Inst. Adv. Study, Princeton, 1965, http://www.math.ku.dk/~grubb/LH65.pdf.
  • L. H örmander, The analysis of linear partial differential operators, I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 256, Springer, Berlin, 1983.
  • L. H örmander, The analysis of linear partial differential operators, III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 274, Springer, Berlin, 1985.
  • L. H örmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications 26, Springer, Berlin, 1997.
  • J. Johnsen, “Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel–Lizorkin spaces”, Math. Scand. 79:1 (1996), 25–85.
  • R. B. Melrose, The Atiyah–Patodi–Singer index theorem, Research Notes in Mathematics 4, A K Peters, Ltd., Wellesley, MA, 1993.
  • L. Boutet de Monvel, “Boundary problems for pseudo-differential operators”, Acta Math. 126:1-2 (1971), 11–51.
  • S. Rempel and B.-W. Schulze, Index theory of elliptic boundary problems, Akademie-Verlag, Berlin, 1982.
  • S. Rempel and B.-W. Schulze, “Complex powers for pseudodifferential boundary problems, II”, Math. Nachr. 116 (1984), 269–314.
  • X. Ros-Oton and J. Serra, “Boundary regularity for fully nonlinear integro-differential equations”, preprint, 2014,.
  • X. Ros-Oton and J. Serra, “The Dirichlet problem for the fractional Laplacian: regularity up to the boundary”, J. Math. Pures Appl. $(9)$ 101:3 (2014), 275–302.
  • E. Schrohe, “A short introduction to Boutet de Monvel's calculus”, pp. 85–116 in Approaches to Singular Analysis, edited by J. Gil et al., Operator Theory: Advances and Applications 125, Birkhäuser Basel, 2001.
  • E. Shargorodsky, “An $L_p$-analogue of the Vishik–Eskin theory”, Memoirs on Differential Equations and Mathematical Physics 2 (1994), 41–146.
  • M. E. Taylor, Pseudodifferential operators, Princeton Mathematical Series 34, Princeton University Press, 1981.
  • H. Triebel, Interpolation theory, function spaces, differential operators, 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995.