On the definition and properties of p-superharmonic functions. (English) Zbl 0572.31004

Lindqvist, Peter

J. Reine Angew. Math. 365, 67-79 (1986).

The regularity and integrability for p-superharmonic functions are investigated. The methods are based on an approximation with regular p- superharmonic functions, which are constructed as solutions of an obstacle problem in the calculus of variations.
In a nonlinear potential theory associated with the p-harmonic equation \[ (1)\quad div(| \nabla u|^{p-2} \nabla u)=0,\quad 1<p<\infty, \] p-superharmonic functions play an important role. Especially, the method of O. Perron can be applied even in the non-linear situation \(p\neq 2\) [cf. S. Granlund, the author and O. Martio, Note on the PWB-method in the nonlinear case (to appear in Pac. J. Math.)]. A function \(v: G\to (-\infty,+\infty]\) is called p-superharmonic in the domain G in \(R^ n\), if (i) v is lower semi-continuous in G, (ii) \(v\not\equiv +\infty\), and (iii) v obeys the comparison principle with respect to p-harmonic functions \((=\) solutions for (1)).
A central result is the following theorem: If v is p-superharmonic in G, \(p>2-1/n\), then the Sobolev derivative \(\nabla v\) exists and \(\int_{D}| \nabla v|^ q dm<\infty\), whenever \(D\subset \subset G\) and \(0<q<n(p-1)/(n-1)\). - Moreover, it is proved that \(v(x)=\lim_{y\to x}ess \inf v(y)\) for each \(x\in G\). An ACL- investigation is included. The methods work even for more general elliptic equations div\((\bar A(x,\nabla u(x)))=0\).

MSC:

31C05	Harmonic, subharmonic, superharmonic functions on other spaces
31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions
35A15	Variational methods applied to PDEs
49J10	Existence theories for free problems in two or more independent variables
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

regularity; integrability; p-superharmonic functions; approximation; nonlinear potential theory; p-harmonic equation; comparison principle; Sobolev derivative