The regularity of solutions to some variational problems, including the $p$-Laplace equation for $2\le p<3$

Cellina, Arrigo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1543-1553

We consider the higher differentiability of solutions to the problem of minimizing

$${\int }_{\Omega }[L\left(\nabla v\left(x\right)\right)+g(x,v\left(x\right))]\mathrm{d}x\text{on}{u}^0+W_0^{1,p}\left(\Omega \right)$$

where L(|ξ|)=1/p|ξ|^p and $u^0\in W^{1,p}\left(\Omega \right)$. We show that, for $2\le p<3$, under suitable regularity assumptions on $g$, there exists a solution $u$ to the Euler–Lagrange equation associated to the minimization problem, such that

$$\nabla u\in W_{\mathrm{loc}}^{1,2}\left(\right).$$

In particular, for $g(x,u)=f\left(x\right)u$ with $f\in W^{1,2}\left(\Omega \right)$ and $2\le p<3$, any $W^{1,p}\left(\Omega \right)$ weak solution to the equation

$$\mathrm{div}\left(\right|\nabla u{|}^{p-2}\nabla u)=f$$

is in $W^{2,2}$${}_{loc}\left(\Omega \right).$

Classification : 49K10
Keywords: Regularity of solutions to variational problems – p-harmonic functions – higher differentiability

B. Avelin and K. Nyström, Estimates for solutions to equations of p-Laplace type in Ahlfors regular NTA-domains. J. Funct. Anal. 266 (2014) 5955–6005. | MR | Zbl | DOI

A. Björn, A regularity classification of boundary points for p-harmonic functions and quasiminimizers. J. Math. Anal. Appl. 338 (2008) 39–47. | MR | Zbl | DOI

L. Damascelli, Comparison Theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Inst. Henri Poincaré, Anal. Non Linéaire 15 (1998) 493–516. | MR | Zbl | Numdam | DOI

E.N. Dancer D. Daners and D. Hauer, A Liouville theorem for p-harmonic functions on exterior domains. Positivity 19 (2015) 577–586. | MR | Zbl | DOI

D. Daners and P. Drábek, A priori estimates for a class of quasi-linear elliptic equations. Trans. Amer. Math. Soc. 361 (2009) 6475–6500. | Zbl | MR | DOI

L. Esposito and G. Mingione, Some remarks on the regularity of weak solutions of degenerate elliptic systems. Rev. Mat. Complutense 11 (1998) 203–219. | Zbl | MR

E. Giusti, Metodi diretti nel calcolo delle variazioni. Unione Matematica Italiana, Bologna (1994). | Zbl | MR

S. Hildebrandt and K.-O. Widman, Some regularity results for quasilinear elliptic systems of second order. Math. Z. 142 (1975) 67–86. | Zbl | MR | DOI

T. Kuusi and G. Mingione, Guide to nonlinear potential estimates. Bull. Math. Sci. 4 (2014) 1–82. | Zbl | MR | DOI

O.A. Ladyzhenskaya and N.N. Uraltseva, Linear and quasilinear elliptic equations. Translated from the Russian. Academic Press, New York, London (1968). | Zbl

P. Lindqvist, Notes on the $p$-Laplace equation, Lecture Notes at the University of Jyvaskyla. Department of Mathematics and Statistics (2006). | Zbl | MR

J.J. Manfredi, A.M. Oberman and A.P. Sviridov, Nonlinear elliptic partial differential equations and p-harmonic functions on graphs. Differ. Integral Equ. 28 (2015) 79–102. | MR | Zbl

L. Nirenberg, Estimates and existence of solutions of elliptic equations. Commun. Pure Appl. Math. 9 (1956) 509–529. | Zbl | MR | DOI

S. Pigola and G. Veronelli, On the Dirichlet problem for p-harmonic maps I: compact targets. Geom. Dedicata 177 (2015) 307–322. | Zbl | MR | DOI

P. Tolksdorf, Regularity for a More General Class of Quasilinear Elliptic Equations. J. Differ. Equ. 51 (1984) 126–150. | Zbl | MR | DOI

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems. Acta Math. 138 (1977) 219–240. | Zbl | MR | DOI