International Journal of Differential Equations

Mixed Boundary Value Problem on Hypersurfaces

Abstract

The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equation ${\text{div}}_{\mathcal{C}}(A{\nabla }_{\mathcal{C}}\phi )=f$ on a smooth hypersurface $\mathcal{C}$ with the boundary $\mathrm{\Gamma }=\partial \mathcal{C}$ in ${\mathbb{R}}^{n}$. $A(x)$ is an $n\timesn$ bounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected parts $\mathrm{\Gamma }={\mathrm{\Gamma }}_{D}\cup {\mathrm{\Gamma }}_{N}$ and on ${\mathrm{\Gamma }}_{D}$ the Dirichlet boundary conditions are prescribed, while on ${\mathrm{\Gamma }}_{N}$ the Neumann conditions. The unique solvability of the mixed BVP is proved, based upon the Green formulae and Lax-Milgram Lemma. Further, the existence of the fundamental solution to ${\text{div}}_{\mathcal{S}}(A{\nabla }_{\mathcal{S}})$ is proved, which is interpreted as the invertibility of this operator in the setting ${\mathbb{H}}_{p,#}^{s}(\mathcal{S})\to {\mathbb{H}}_{p,#}^{s-2}(\mathcal{S})$, where ${\mathbb{H}}_{p,#}^{s}(\mathcal{S})$ is a subspace of the Bessel potential space and consists of functions with mean value zero.

Article information

Source
Int. J. Differ. Equ., Volume 2014 (2014), Article ID 245350, 8 pages.

Dates
Revised: 31 May 2014
Accepted: 2 June 2014
First available in Project Euclid: 20 January 2017

https://projecteuclid.org/euclid.ijde/1484881410

Digital Object Identifier
doi:10.1155/2014/245350

Mathematical Reviews number (MathSciNet)
MR3253609

Zentralblatt MATH identifier
1308.35116

Citation

DuDuchava, R.; Tsaava, M.; Tsutsunava, T. Mixed Boundary Value Problem on Hypersurfaces. Int. J. Differ. Equ. 2014 (2014), Article ID 245350, 8 pages. doi:10.1155/2014/245350. https://projecteuclid.org/euclid.ijde/1484881410

References

• R. Duduchava, “The Green formula and layer potentials,” Integral Equations and Operator Theory, vol. 41, no. 2, pp. 127–178, 2001.MR1847170
• R. Duduchava, D. Mitrea, and M. Mitrea, “Differential operators and boundary value problems on hypersurfaces,” Mathematische Nachrichten, vol. 279, no. 9-10, pp. 996–1023, 2006.
• R. Duduchava, “Partial differential equations on hypersurfaces,” Georgian Academy of Sciences A: Memoirs on Differential Equations and Mathematical Physics, vol. 48, pp. 19–74, 2009.MR2603279
• M. Mitrea and M. Taylor, “Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem,” Journal of Functional Analysis, vol. 176, no. 1, pp. 1–79, 2000.MR1781631
• H. Le Dret, Numerical Approximation of PDEs, Class Notes M1 Mathematics, 2011-2012, 2011, http://www.ann.jussieu.fr/ledret/M1ApproxPDE.html.
• W. Haack, Elementare Differentialgeometrie (Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften), Birkhäauser, Basel, Switzerland, 1955, (German).
• R. Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Prentice-Hall, Englewood Cliffs, NJ, USA, 1962.
• P. G. Ciarlet, Introduction to Linear Shell Theory, vol. 1 of Series in Applied Mathematics, Gauthier-Villars, Paris, France, 1998.
• L. Andersson and P. T. Chruściel, “Hyperboloidal Cauchy data for vacuum Einstein equations and obstructions to smoothness of null infinity,” Physical Review Letters, vol. 70, no. 19, pp. 2829–2832, 1993.MR1215407
• R. Temam and M. Ziane, “Navier-stokes equations in thin spherical domains,” in Optimization Methods in Partial Differential Equations, vol. 209 of Contemporaty Math., pp. 281–314, AMS, 1996.
• R. Duduchava, “A revised asymptotic model of a shell,” Memoirs on Differential Equations and Mathematical Physics, vol. 52, pp. 65–108, 2011.MR2883795
• H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Johann Ambrosius Barth, Heidelberg, Germany, 2nd edition, 1995.MR1328645
• L. Hörmander, The Analysis of Linear Partial Differential Operators IV, Springer, Heidelberg, Germany, 1983.
• G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, vol. 164 of Applied Mathematical Sciences Series, Springer, Berlin, Germany, 2008.
• M. Shubin, Pseudodifferential Operators and Spectral Theory, Springer, Berlin, Germany, 1987.
• M. E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, USA, 1981.MR618463
• R. Duduchava, “On Noether theorems for singular integral equations,” in Proceedings of the Symposium on Mechanics and Related Problems of Analysis, vol. 1, pp. 19–52, Metsniereba, Tbilisi, Georgia, 1973, (Russian).
• P. D. Lax and A. N. Milgram, “Parabolic equations,” in Contributions to the theory of partial differential equations, Annals of Mathematics Studies, pp. 167–190, Princet on University Press, Princeton, NJ, USA, 1954.MR0067317
• J. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Springer, Heidelberg, Germany, 1972.MR0350177 \endinput