Hypoellipticity for infinitely degenerate quasilinear equations and the Dirichlet problem. (English) Zbl 1310.35081

Summary: In [C. Rios et al., Differ. Integral Equ. 21, No. 1–2, 131–200 (2008; Zbl 1224.35149)], we considered a class of infinitely degenerate quasilinear equations of the form div \(A(x,\omega)\nabla\omega+\overrightarrow\gamma(x,\omega)\cdot\nabla\omega+f(x,\omega)=0\) and derived a priori bounds for high order derivatives \(D^\alpha\omega\) of their solutions in terms of \(\omega\) and \(\omega\). We now show that it is possible to obtain bounds in terms of just \(\omega\) for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness, and interior \(\varrho^\infty\)-regularity of solutions for the corresponding Dirichlet problem with continuous boundary data.


35H10 Hypoelliptic equations
35J62 Quasilinear elliptic equations
35J70 Degenerate elliptic equations
35D35 Strong solutions to PDEs


Zbl 1224.35149
Full Text: DOI arXiv


[1] R. Bouldin, The norm continuity properties of square roots, SIAM J. Math. Anal. 4 (1972), 206–210. · Zbl 0241.47019
[2] L. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998. · Zbl 0902.35002
[3] V. S. Fediĭ, On a criterion for hypoellipticity, Math. USSR Sb. 14 (1971), 15–45. · Zbl 0247.35023
[4] B. Franchi, R. Serapioni, and F. Serra Cassano, Meyer-Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math. 22 (1996), 859–890. · Zbl 0876.49014
[5] N. Garofalo and D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), 1081–1144. · Zbl 0880.35032
[6] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, revised 3rd printing, Springer-Verlag, 1998.
[7] J. J. Kohn, Hypoellipticity of some degenerate subelliptic operators, J. Funct. Anal. 159 (1998), 203–216. · Zbl 0937.35024
[8] S. Kusuoka and D. Stroock, Applications of the Malliavin calculus II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), 1–76. · Zbl 0568.60059
[9] C. Rios, E. Sawyer, and R. Wheeden, A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations, Adv. in Math. 193 (2005), 373–415. · Zbl 1072.35052
[10] C. Rios, E. Sawyer and R. Wheeden, A priori estimates for infinitely degenerate quasilinear equations, Differential Integral Equations 21 (2008), 131–200. · Zbl 1224.35149
[11] E. Sawyer and R. Wheeden, Regularity of degenerate Monge-Ampère and prescribed Gaussian curvature equations in two dimensions, Potential Anal. 24 (2006), 267–301. · Zbl 1097.35035
[12] E. Sawyer and R. Wheeden, A priori estimates for quasilinear equations related to the Monge-Ampère equation in two dimensions, J. Anal. Math. 97 (2005), 257–316. · Zbl 1260.35052
[13] E. Sawyer and R. Wheeden, Degenerate Sobolev spaces and regularity of subelliptic equations, Trans. Amer. Math. Soc. 362 (2009), 1869–1906. · Zbl 1191.35085
[14] F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1, Pseudodifferential Operators, Plenum Press, New York – London, 1980.
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