## 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.

### MSC:

 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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### References:

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