Lieberman, Gary M. The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. (English) Zbl 0742.35028 Commun. Partial Differ. Equations 16, No. 2-3, 311-361 (1991). Elliptic quasilinear differential equations of the type \(Qu=\hbox{div}(A(x,u,Du))+B(x,u,Du)\) are the subject of the classical book of O. A. Ladyzhenskaya and N. N. Ural’tseva, “Linear and quasilinear elliptic equations” (1968; Zbl 0164.13002),(1964; Zbl 0143.33602). For the purposes of this review we will suppose that \(B=0\) and that \(A\) depends only on \(x\) and \(Du\). To prove the boundedness of weak solutions the condition considered by Ladyzhenskaya and Ural’tseva is \(A(x,z,p) p\sim| p|^ m\) for some \(m>1\). To obtain regularity of the derivatives of the solutions they need to assume, in addition, that \(A\) is differentiable and that \(a_{i,j}=\partial A^ i/\partial p_ j\) has eigenvalues proportional to \((1+| p|)^{m-2}\) and \(\partial A/\partial x\preceq| p|^ m\). Under these assumptions \(Du\) turns out to be Hölder continuous.The prototype equation is \(\hbox{div}((1+| Du|^ 2)^{(m- 2)/2}Du)=0\).As indicated in the paper under review, through the work of several authors these results were extended to cover the degenerate case, where we only require that the ratio of the largest to the smallest eigenvalue of \(a_{i,j}\) is bounded independently of \(Du\). The prototype equation is then the \(m\)-Laplacian \(\hbox{div}(| Du|^{(m-2)}Du)=0\). Consider the most general degenerate equation, where we further restrict \(A\) to depend only on \(| Du|\). We can write it as \(\hbox{div}(g(| Du|)Du/| Du|)=0\), for some function \(g\). This equation is elliptic if and only if there are positive constants \(\delta_ 1\) and \(\delta_ 2\) such that for \(t>0\), \(\delta_ 1\leq tg'(t)/g(t)\leq\delta_ 2\).In this paper the author extends the Ladyzhenskaya-Ural’tseva estimates when \(| p|^ m\) is replaced by \(| p| g(| p|)\), where \(g\) satisfies the above condition, which is the ‘natural’ extension of the natural conditions of Ladyzhenskaya and Ural’tseva. This extension is very significant since it allows to consider functions \(g(t)\) that behave like \(t^ \alpha\) for \(t\to 0\) and like \(t^ \beta\) for \(t\to\infty\), where \(0<\alpha\leq\beta\). In addition, the proofs are technically more involved, since one can no longer take advantage of the homogeneity of \(t^ m\). The natural Sobolev spaces are defined by means of \(G(t)=\int^ t_ 0g(s)ds\) as follows. The space \(W^{1,G}(\Omega)\) consists of weakly differentiable functions such that \(\int_ \Omega G(| Du|)dx<\infty\).In the paper the full nonhomogeneous equation is treated and both interior and boundary regularity are considered. The definition of De Giorgi classes is generalized to take into account \(G\) and regularity and Harnack inequality results are extended to these new classes. Reviewer: J.J.Manfredi (Pittsburgh) Cited in 7 ReviewsCited in 545 Documents MSC: 35J70 Degenerate elliptic equations 35D10 Regularity of generalized solutions of PDE (MSC2000) 35J60 Nonlinear elliptic equations 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:interior regularity; natural growth condition; Sobolev-Orlicz spaces; Elliptic quasilinear differential equations; Hölder continuous; \(m\)-Laplacian; boundary regularity; De Giorgi classes; Harnack inequality Citations:Zbl 0164.13002; Zbl 0143.33602 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] DeGiorgi E., Mem. Accad. Sci. Torino Ci. Sci. Fis. Mat. Natur 33 pp 25– (1957) [2] DiBenedetto E., Nonlinear Anal. 7 pp 827– (1983) · Zbl 0539.35027 · doi:10.1016/0362-546X(83)90061-5 [3] DiBenedetto E., Ann Inst. pp 295– (1984) [4] Donaldson T. K., J. Funct. 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