On the definition and properties of p-superharmonic functions.

*(English)*Zbl 0572.31004The 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\((\bar A(x,\nabla u(x)))=0\).

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 |