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The \(p\)-harmonic system with measure-valued right hand side. (English) Zbl 0879.35052
For \(2-1/n<p<n\) one proves existence of a distributional solution \(u\) of the \(p\)-harmonic system \[ -\text{div}(|\nabla u|^{p-2}\nabla u)= \mu\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega, \] where \(\Omega\) is an open subset of \(\mathbb{R}^n\) (bounded or unbounded), \(u:\Omega\to\mathbb{R}^n\), and \(\mu\) is an \(\mathbb{R}^m\)-valued Radon measure of finite mass. For the solution \(u\) one establishes the Lorentz space estimate \[ |Du|_{Lq,\infty}+|u|_{Lq^*,\infty}\leq C|\mu|M_{1/(p-1)}, \] \(q=n/(n-1)(p-1)\) and \(q^*=n/(n-p)(p-1)\). The main step of the proof is to show that for suitable approximations the gradients \(Du_k\) converge. This is achieved by the choice of regularized test functions and a localization argument to compensate the fact that in general \(u\not\in W_{1,p}\).

MSC:
35J60 Nonlinear elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
35D05 Existence of generalized solutions of PDE (MSC2000)
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References:
[1] Bénilan, P.; Boccardo, L.; Gallouët, T.; Gariepy, R.; Pierre, M.; Vazquez, J.L., An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. scuola norm. sup. Pisa cl. sci., Vol. 22, 4, 241-273, (1995) · Zbl 0866.35037
[2] Boccardo, L.; Gallouët, T., Nonlinear elliptic and parabolic equations involving measure data, J. funct. anal., Vol. 87, 149-169, (1989) · Zbl 0707.35060
[3] Boccardo, L.; Gallouët, T., Nonlinear elliptic equations with right-hand side measures, Comm. part. diff. eqn., Vol. 17, 641-655, (1992) · Zbl 0812.35043
[4] Boccardo, L.; Murat, F., Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear anal., Vol. 19, 581-597, (1992) · Zbl 0783.35020
[5] Chen, Y.; Hong, M.-C.; Hungerbühler, N., Heat flow of p-harmonic maps with values into spheres, Math. Z., Vol. 215, 25-35, (1994) · Zbl 0793.53049
[6] Dibenedetto, E.; Manfredi, J., On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Am. J. math., Vol. 115, 1107-1134, (1993) · Zbl 0805.35037
[7] Evans, L.C., Weak convergence methods for nonlinear partial differential equations, (), No. 74 · Zbl 0349.34043
[8] Evans, L.C.; Gariepy, R., Measure theory and fine properties of functions, (1992), CRC Press Boca Raton, (etc.) · Zbl 0804.28001
[9] Fuchs, M., The Green-matrix for elliptic systems which satisfy the Legendre-Hadamard condition, Manuscr. math., Vol. 46, 97-115, (1984) · Zbl 0552.35025
[10] Fuchs, M., The Green matrix for strongly elliptic systems of second order with continuous coefficients, Z. anal. anwend., Vol. 5, 6, 507-531, (1986) · Zbl 0634.35023
[11] Fuchs, M., The blow-up of p-harmonic maps, Manuscr. math., Vol. 81, 89-94, (1993) · Zbl 0794.58012
[12] Fuchs, M.; Reuling, J., Non-linear elliptic systems involving measure data, Rend. math. appl., Vol. 15, 311-319, (1995) · Zbl 0838.35133
[13] Grüter, M.; Widman, K.-O., The Green function for uniformly elliptic equations, Manuscr. math., Vol. 37, 303-342, (1982) · Zbl 0485.35031
[14] {\scP. L. Lions} and {\scF. Murat}, Solutions renormalisées d’equations elliptiques (to appear).
[15] Müller, S., Det = det. A remark on the distributional determinant, C.R. acad. sci., Paris, Sér. I, Vol. 311, 13-17, (1990) · Zbl 0717.46033
[16] {\scF. Murat}, Équations elliptiques non linéaires avec second membre L1 ou mesure. Preprint.
[17] Murat, F., Soluciones renormalizadas de EDP elipticas no lineales, ()
[18] Rakotoson, J.M., Generalized solutions in a new type of sets for problems with measure data, Differ. integral eqn., Vol. 6, 27-36, (1993) · Zbl 0780.35047
[19] Talenti, G., Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. mat. pura appl., IV. ser., Vol. 120, 159-184, (1979) · Zbl 0419.35041
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