×

Jump discontinuous viscosity solutions to second order degenerate elliptic equations. (English) Zbl 1351.35036

The authors study the Dirichlet problem for linear degenerate elliptic partial differential equations. They interpret solutions in the viscosity sense, and assume the equation is only degenerate in the normal direction along a strictly interior \(n-1\) dimensional closed surface that divides the domain into two components. It was already known that viscosity solutions can be discontinuous along the surface, due to the degeneracy. If the equation is strictly monotone in the zeroth order term, the authors show that the PDE can be decoupled into two PDEs. One PDE holds inside and on the closed surface (with no boundary data), and the other PDE holds outside the closed surface with the original Dirichlet conditions. The solutions of the two independent PDEs are equal on the separating surface, and when pieced together give the unique viscosity solution of the original PDE. The authors also show that the value of the solution along the separating surface can be computed by solving the original PDE restricted to the surface. When the equation is not strictly monotone in the zeroth order term, the solution may be discontinuous along the separating surface. In this case the authors show that the decomposition still holds, and they obtain the unique discontinuous viscosity solution of the PDE, up to an ambiguity in the assignment of the values of the solution along the surface of discontinuity. The authors also examine gradient blow-up of solutions near the separating surface.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35D40 Viscosity solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J70 Degenerate elliptic equations
35B44 Blow-up in context of PDEs
35B51 Comparison principles in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bardi, M.; Capuzzo Dolcetta, I., Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (1997), Birkhäuser: Birkhäuser Boston · Zbl 0890.49011
[2] Bardi, M.; Mannucci, P., On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations, Commun. Pure Appl. Anal., 5, 709-731 (2006) · Zbl 1142.35041
[3] Barles, G., Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit, Nonlinear Anal., 20, 1123-1134 (1993) · Zbl 0816.35081
[4] Barles, G., Solutions de Viscositè des Équations de Hamilton-Jacobi (1994), Springer-Verlag: Springer-Verlag Berlin · Zbl 0819.35002
[5] Bernstein, S., Sur la generalisation du Probleleme de Dirichlet, Part I, Math. Ann.. Math. Ann., Math. Ann., 69, 82-136 (1910) · JFM 41.0427.02
[6] Bernstein, S., Conditions nècessaires et suffisantes pour la posibilité du problème de Dirichlet, C. R. Acad. Sci. Paris, 150, 514-515 (1910) · JFM 41.0427.01
[7] Black, F.; Scholes, M., The valuation of options and corporate liabilities, J. Political Economy, 81, 637-654 (1973) · Zbl 1092.91524
[8] Chobanov, G.; Kutev, N., Interior boundaries for linear degenerate elliptic equations, Mediterr. J. Math., 9, 789-801 (2012) · Zbl 1257.35098
[9] Crandall, M.; Ishii, H.; Lionis, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27, 1-67 (1992) · Zbl 0755.35015
[11] Fichera, G., On a unified theory of boundary value problem for elliptic-parabolic equations of second order, (Boundary Problems in Differential Equations (1960), Univ. of Wisc. Press: Univ. of Wisc. Press Madison), 97-120
[12] Huan, Z., Semi-jets on boundaries and well-posedness of boundary value problems, Appl. Anal., 67, 341-356 (1997) · Zbl 0894.35039
[13] Kawohl, B.; Kutev, N., Strong maximum principle for semicontinuous viscosity solutions of nonlinear partial differential equations, Arch. Math., 70, 470-478 (1998) · Zbl 0907.35008
[14] Kawohl, B.; Kutev, N., Viscosity solutions for degenerate and nonmonotone elliptic equations, (Sequeira, A.; etal., Applied Nonlinear Analysis (1999), Kluwer/Plenum: Kluwer/Plenum New York), 231-254 · Zbl 0960.35040
[15] Kawohl, B.; Kutev, N., Comparison principle for viscosity solutions of fully nonlinear degenerate elliptic equations, Comm. Partial Differential Equations, 32, 1209-1224 (2007) · Zbl 1185.35084
[16] Kawohl, B.; Kutev, N., A study on the gradient blow up for viscosity solutions of fully nonlinear, uniformly elliptic equations, Acta Math. Sci., 32, 15-40 (2012) · Zbl 1265.35076
[17] Lee, Chung-Min; Rubinstein, J., Elliptic equations with diffusion coefficient vanishing at the boundary: theoretical and computational aspects, Quart. Appl. Math., 64, 4, 735-747 (2006) · Zbl 1141.35382
[18] Oleinik, O. A.; Radkevich, E. V., Second Order Equations with Nonnegative Characteristic Form (1971), Itogi Nauki: Itogi Nauki Moscow · Zbl 0217.41502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.