On the Regularity Solutions to a Class of p-Harmonic Equations

Gioconda MOSCARIELLO Dipartimento di Matematica e Applicazioni ”R. Caccioppoli” Università di Napoli “Federico II”, via Cintia - 80126 Napoli

                  e-mail: e-mail:gmoscari@unina.it

Key words: Elliptic equations; Mappings with finite distortion.

AMS Classifications: 35B45; 35J60; 30C65.

Abstract. We present local estimates and regularity results for the gradient of solutions to some degenerate elliptic PDEs whose model arises in the study of mappings with finite distortion. More precisely, we consider equations of the type

(1)  div⟨H(x)Du, Du⟩^{\frac{p−2}{2}} H(x)Du = 0  in Ω ⊂ ℝ^{n}

where H(x) : ℝ^{m×n} → ℝ^{m×n} is a symmetric positive linear transformation of ℝ^{m×n} and 1 ≤ m, 2 ≤ n, 2 ≤ p. Following a common use in the framework of the mappings with finite distortion (see e. g. [6], [5], [4]) we will express ellipticity condition by the inequality

(2)  \frac{|ξ|^{p}}{p} + \frac{|H(x, ξ)|^{q}}{q} ≤ K(x)⟨H(x, ξ), ξ⟩

where pq = p + q and H(x, ξ) = ⟨H(x)ξ, ξ⟩^{\frac{p−2}{2}} H(x)ξ for almost every x ∈ Ω and all vectors ξ ∈ ℝ^{m×n}.

The nonnegative function K(x), which measures the degree of degeneracy of ellipticity bounds lies in the exponential class EXP (Ω), that is

(3)  ∫_{Ω} e^{λK}dx < ∞

for some λ > 0.

Here we deal with solutions of (1) having ”finite energy”, namely

E[u] = ∫_{Ω}⟨H(x)Du, Du⟩^{\frac{p}{2}} < ∞

By assumptions (2) and (3), the gradient of such solutions lies in the Orlicz-Zygmund space L^{p} log^{−1} L(Ω, ℝ^{m×n}). In [2] an a priori estimate is established for solutions having gradient in L^{p} log^{α} L, α ≥ 0. The precise statement is the following

Theorem For every α ≥ 0 there exists the so-called critical exponent λ_{α} = λ_{α}(n, p) such that if

(3)  ∫_{Ω} e^{λK}dx < ∞

for some λ > λ_{α} then every solution of (1) satisfies the following uniform bound

||Du||_{L^{p} log^{α} L(Ω′)}^{p} ≤ C(Ω′)E[u]

for each compact Ω′ ⊂ Ω.

In case p = 2 or p = m = n every finite energy solution has locally L^{p} log^{α} L integrable gradient ([2], [1]) .

References

[1] T. Iwaniec, P. Koskela, G. Martin, C. Sbordone, Mappings of exponentially integrable distortion, to appear on London Math. J.

[2] T. Iwaniec, L. Migliaccio, G. Moscariello, A. Passarelli di Napoli, A priori estimates for nonlinear elliptic complexes, preprint (2000).

[3] T. Iwaniec, C. Sbordone, Quasiharmonic fields, Ann. Inst. H. Poincaré, AN 18 (2001), 519-572.

[4] J. Manfredi, Weakly monotone functions, The Journal of Geometric Anal., 4 (1994), 393-402.

[5] J. Manfredi, E. Villamor, An extension of Reshetnyak’s theorem. Indiana Univ. Math. J. 47, 3 (1998), 1131-1145.

[6] L. Migliaccio, G. Moscariello , Higher integrability of div-curl products, Ricerche di Matematica, vol. XLIX, (2000), 151-161.