Inverting the \(p\)-harmonic operator. (English) Zbl 0869.35037

Summary: The paper is concerned with the nonhomogeneous \(p\)-harmonic equation \[ \text{div}|\nabla u|^{p-2}\nabla u=\text{div }f. \] The main object is the operator \(\mathcal H\) which carries the given vector function \(f\) into the gradient field \(\nabla u\). The natural domain of \(\mathcal H\) is the Lebesgue space \(L^q(\Omega,\mathbb{R}^n)\), where \(1/p+ 1/q=1\). We extend the operator \(\mathcal H\) to slightly larger spaces called grand \(L^q\)-spaces. Continuity of \(\mathcal H\) is used to prove existence and uniqueness results for nonhomogeneous \(n\)-harmonic type equations \(\text{div }{\mathcal A}(x,\nabla u)=\mu\), with \(\mu\) a Radon measure.


35J60 Nonlinear elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
47J25 Iterative procedures involving nonlinear operators
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[1] Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vasquez, J. L.: AnL 1-Theory of Existence and Uniqueness of Solutions of Nonlinear Elliptic Equations, Ann. Scuola Norm. Sup. Pisa, Ser. IV,22 (1995), 241–273. · Zbl 0866.35037
[2] Boccardo, L., Gallouët, T.: Non linear elliptic and parabolic equations involving measure data, Jour. Func. Anal.,87 (1989), 149–169. · Zbl 0707.35060
[3] Brezis, H., Merle, F.: Uniform estimates and blow-up behavior for solutions of {\(\Delta\)}u =V(x)e u in two dimensions, Comm. In P.D.E.16 (1991), 1223–1253. · Zbl 0746.35006
[4] Chanillo, S., Li, Y. Y.: Continuity of solutions of uniformly elliptic equations in \(\mathbb{R}\)2, Manuscripta Math.,77, n. 4 (1992), 415–433. · Zbl 0797.35031
[5] Dolzman, G., Hungerbühler, N., Müller, S.: Nonlinear elliptic systems with measure-valued right hand side, to appear. · Zbl 0895.35029
[6] Fiorenza, A., Sbordone, C.: Existence and uniqueness results for solutions of nonlinear equations with right hand side inL 1, to appear. · Zbl 0891.35039
[7] Greco, L.: A remark on the equality detD f = DetD f, Diff. Int. Eq.,6, n. 5, (1993), 1089–1100. · Zbl 0784.49013
[8] Greco, L., Iwaniec, T., Sbordone, C., Stroffolini, B.: Degree formulas for maps with nonintegrable jacobian, Topological Methods in Nonlinear Analysis,6, n. 1 (1995) 81–95. · Zbl 0854.58005
[9] Iwaniec, T.:p-Harmonic tensors and quasiregular mappings, Annals of Math,136 (1992), 589–624. · Zbl 0785.30009
[10] Iwaniec, T. Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses, Arch. Rat. Mech. Anal.119 (1992), 129–143. · Zbl 0766.46016
[11] Iwaniec, T., Sbordone, C.: Weak minima of variational integrals, J. Reine Angew. Math.454 (1994), 143–161. · Zbl 0802.35016
[12] Iwaniec, T., Scott, C., Stroffolini, B.: Nonlinear Hodge theory on manifolds with boundary, to appear. · Zbl 0963.58003
[13] Lions, P.-L., Murat, F.: Solutions renormalisées d’equations elliptiques, to appear.
[14] Kilpeläinen, T., Malý, J.: Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup; Pisa, Ser IV,19 (1992), 591–613. · Zbl 0797.35052
[15] Murat, F.: Soluciones renormalizadas de EDP elipticas no lineales, Publications du Laboratoire d’Analyse Numerique, Paris (1993).
[16] Stampacchia, G.: Le problème de Dirichlet pour les equations elliptiques du second ordrea coefficients discontinus, Ann. Inst. Fourier, Grenoble,15 (1965) 189–258. · Zbl 0151.15401
[17] Del Vecchio, T.: Nonlinear elliptic equations with measure data, Potential Analysis4 (1995), 185–203. · Zbl 0815.35023
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