Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems with Rough Coefficients

Rodney, Scott

Canadian journal of mathematics, Tome 64 (2012) no. 6, pp. 1395-1414

Résumé

This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form $${\nabla }'P\left( x \right)\nabla u+\text{HR}u+\text{{S}'G}u\text{+ F}u\text{=}\text{f}\,\text{+}\,\text{{T}'g}\,\text{in}\,\Theta$$ $$u=\varphi \,\,on\,\,\partial \Theta$$ The principal part ${\xi }'P\left( x \right)\xi$ of the above equation is assumed to be comparable to a quadratic form $\mathcal{Q}\left( x,\xi\right)={\xi }'Q\left( x \right)\xi$ that may vanish for non-zero $\xi \in {{\mathbb{R}}^{n}}$ . This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $Q{{H}^{1}}\left( \Theta\right)={{W}^{1,2}}\left( \Theta ,Q \right)$ and $QH_{0}^{1}\left( \Theta\right)=W_{0}^{1,2}\left( \Theta ,Q \right)$ as defined in previous works. E.T. Sawyer and R.L. Wheeden (2010) have given a regularity theory for a subset of the class of equations dealt with here.

Détail

DOI : 10.4153/CJM-2012-029-1

Mots-clés : 35A01, 35A02, 35D30, 35J70, 35H20, degenerate quadratic forms, linear equations, rough coefficients, subelliptic, weak solutions.

Bibliographie

[CRW] [CRW] Chua, S. K., Rodney, S., and Wheeden, R. L., A compact embedding theorem for generalized Sobolev spaces. arxiv:1110.6907v1. Google Scholar

[FKS] [FKS] Fabes, E. B., Kenig, C. E., and Serapioni, R. P., The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7(1982), no. 1, 77–116. Google Scholar | DOI

[FP] [FP] Fefferman, C. and Phong, D. H., Subelliptic eigenvalue problems. In: Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983. Google Scholar

[FSSC] [FSSC] Franchi, B., Serapioni, R., and F. Serra-Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields. Boll. Un. Mat. Ital. B. (7) 11(1997), no. 1, 83–117. Google Scholar

[GL] [GL] Gutiérrez, C. E. and Lanconelli, E., Maximum principle, non-homogeneous Harnack inequality, and Liouville theorems for x-elliptic operators. Comm. Partial Differential Equations 28(2003), no. 11–12, 1833–1862. Google Scholar | DOI

[GT] [GT] Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, New York, 1977. Google Scholar

[MRW] [MRW] Monticelli, D. D., Rodney, S., and Wheeden, R. L., Boundedness of weak solutions of degenerate quasilinear equations with rough coefficients. Differential Integral Equations 25(2012), no. 1–2, 143–200. Google Scholar

[R1] [R1] Rodney, S. W., Existence of weak solutions to subelliptic partial differential equations in divergence form and the necessity of the Sobolev and Poincaré inequalities. Ph. D. Thesis, Mc Master University, Proquest LLC, Ann Arbor, MI, 2007. Google Scholar

[R2] [R2] Rodney, S. W., A degenerate Sobolev inequality for a large open set in a homogeneous space. Trans. Amer. Math. Soc. 362(2010), no. 2, 673–685. Google Scholar | DOI

[SW1] [SW1] Sawyer, E. T. and Wheeden, R. L., Hölder continuity of weak solutions to subelliptic equations with rough coefficients. Mem. Amer. Math. Soc. 180(2006), no. 847. Google Scholar

[SW2] [SW2] Sawyer, E. T. and Wheeden, R. L., Degenerate Sobolev spaces and regularity of subelliptic equations. Trans. Amer. Math. Soc. 362(2010), no. 4, 1869–1906. http://dx.doi.org/10.1090/S0002-9947-09-04756-4 Google Scholar