On \(W^{1,p}\)-solvability for special vectorial Hamilton–Jacobi systems. (English) Zbl 1035.35025
In this paper a rather weak notion of almost everywhere solution in a Sobolev space to the Dirichlet problem
\[
F(Du(x))= 0,\quad x\in\Omega,\qquad u(x)= \varphi(x),\quad x\in\partial\Omega,
\]
is studied, where \(\Omega\) is a bounded open set in \(\mathbb{R}^n\) and \(u: \Omega\to\mathbb{R}^m\) is a unknown vector field.
The general existence theorems for certain Dirichlet problems using suitable approximation schemes called \(W^{1,p}\)-reduction principles that generalize the similar reduction principle for Lipschitz solutions are established. The method relies on a new Baire’s category argument concerning the residual continuity of a Baire-one function. Some sufficient conditions for \(W^{1,p}\)-reduction are also given along with certain generalization of some known results and a specific application to the boundary value problem for special weakly quasiregular mappings.
The general existence theorems for certain Dirichlet problems using suitable approximation schemes called \(W^{1,p}\)-reduction principles that generalize the similar reduction principle for Lipschitz solutions are established. The method relies on a new Baire’s category argument concerning the residual continuity of a Baire-one function. Some sufficient conditions for \(W^{1,p}\)-reduction are also given along with certain generalization of some known results and a specific application to the boundary value problem for special weakly quasiregular mappings.
Reviewer: V. Iftode (Bucureşti)
MSC:
| 35F30 | Boundary value problems for nonlinear first-order PDEs |
| 35A25 | Other special methods applied to PDEs |
| 49J10 | Existence theories for free problems in two or more independent variables |
References:
| [1] | Bruckner, A.; Bruckner, J.; Thomson, B., Real Analysis (1997), Prentice-Hall · Zbl 0872.26001 |
| [2] | Cardaliaguet, P.; Dacorogna, B.; Gangbo, W.; Georgy, N., Geometric restrictions for the existence of viscosity solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16, 189-220 (1999) · Zbl 0927.35021 |
| [3] | Crandall, M. G.; Evans, L. C.; Lions, P. L., Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282, 487-502 (1984) · Zbl 0543.35011 |
| [4] | Dacorogna, B.; Marcellini, P., General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math., 178, 1, 1-37 (1997) · Zbl 0901.49027 |
| [5] | Dacorogna, B.; Marcellini, P., Cauchy-Dirichlet problem for first order nonlinear systems, J. Funct. Anal., 152, 2, 404-446 (1998) · Zbl 0911.35034 |
| [6] | Gromov, M., Partial Differential Relations (1986), Springer · Zbl 0651.53001 |
| [7] | Kirchheim, B., Deformations with finitely many gradients and stability of quasiconvex hulls, C. R. Acad. Sci. Paris Ser. I Math., 332, 289-294 (2001) · Zbl 0989.49013 |
| [8] | Lions, P.-L., Generalized Solutions of Hamilton-Jacobi Equations. Generalized Solutions of Hamilton-Jacobi Equations, Pitman Research Notes in Mathematics, 69 (1982), Longman · Zbl 0497.35001 |
| [9] | Müller, S., Variational models for microstructure and phase transitions, (Calculus of Variations and Geometric Evolution Problems, Cetraro, 1996. Calculus of Variations and Geometric Evolution Problems, Cetraro, 1996, Lecture Notes in Math., 1713 (1999), Springer), 85-210 · Zbl 0968.74050 |
| [10] | Müller, S.; Šverák, V., Attainment results for the two-well problem by convex integration, (Geometric Analysis and the Calculus of Variations (1996), Internat. Press), 239-251 · Zbl 0930.35038 |
| [11] | Müller, S.; Šverák, V., Convex integration with constraints and applications to phase transitions and partial differential equations, J. Eur. Math. Soc., 1, 4, 393-422 (1999) · Zbl 0953.35042 |
| [12] | Müller, S.; Sychev, M., Optimal existence theorems for nonhomogeneous differential inclusions, J. Funct. Anal., 181, 2, 447-475 (2001) · Zbl 0989.49012 |
| [13] | Yan, B., A linear boundary value problem for weakly quasiregular mappings in space, Cal. Var. Partial Differential Equations, 13, 295-310 (2001) · Zbl 0992.30009 |
| [14] | Yan, B., Relaxation and attainment results for an integral functional with unbounded energy-well, Proc. Roy. Soc. Edinburgh, 132A, 1513-1523 (2002) · Zbl 1026.49009 |
| [15] | B. Yan, A Baire’s category method for Dirichlet problem of quasiregular mappings, Trans. Amer. Math. Soc., submitted for publication; B. Yan, A Baire’s category method for Dirichlet problem of quasiregular mappings, Trans. Amer. Math. Soc., submitted for publication · Zbl 1029.30015 |
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