A Quantitative Second Order Sobolev Regularity for (inhomogeneous) Normalized p(•)-Laplace Equations

Let Omega be a domain of R-n with n >= 2 and p(center dot) be a local Lipschitz funcion in Omega with 1 < p(x) < infinity in Omega. We build up an interior quantitative second order Sobolev regularity for the normalized p(center dot)-Laplace equation -Delta(N)(p(center dot))u = 0 in Omega as well as the corresponding inhomogeneous equation -Delta(N)(p(center dot))u =f in Omega with f is an element of C-0(Omega). In particular, given any viscosity solution u to -Delta(N)(p(center dot))u = 0 in Omega, we prove the following: (i) in dimension n = 2, for any subdomain U (sic) Omega and any beta >= 0, one has divided by Du divided by beta Du is an element of L(loc)2+delta (U) with a quantitative upper bound, and moreover, the map (x(1),x(2))->|Du|(beta)(u(x1),-u(x2)) is quasiregular in U in the sense that |D[|Du|(beta)Du]|(2)<=-CdetD[|Du|(beta)Du]a.e.inU. (ii)in dimension n >= 3, for any subdomain U (sic) Omega with inf(U) p(x) > 1 and supUp(x)<3+2n-2, one has D(2)u is an element of L-loc(2+delta) (U) with a quantitative upper bound, and also with a pointwise upper bound |D(2)u|(2)<=-C Sigma(1 <= i<j <= n)[u(xixj)u(xjxi)-u(xixi)u(xjxj)]a.e in U. Here constants delta > 0 and C >= 1 are independent of u. These extend the related results obtaind by Adamowicz-H & auml;st & ouml; [Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, 10, 1940-1965 (2010)] when n = 2 and beta = 0.