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173
Regularity of minima: An invitation to the dark SIDE OF THE CALCULUS OF VARIATIONS
, 2006
"... I am presenting a survey of regularity results for both minima of variational integrals, and solutions to nonlinear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to ..."
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Cited by 41 (4 self)
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I am presenting a survey of regularity results for both minima of variational integrals, and solutions to nonlinear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to
On the weak continuity of elliptic operators and applications to potential theory
 Amer. J. Math
"... On the weak continuity of elliptic operators and applications to potential theory ..."
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Cited by 39 (7 self)
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On the weak continuity of elliptic operators and applications to potential theory
QUASILINEAR AND HESSIAN EQUATIONS OF LANE–EMDEN TYPE
, 2005
"... The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane–Emden type, including the following two model problems: −∆pu = u q + µ, Fk[−u] = u q + µ, u ≥ 0, on R n, or on a bounded domain Ω ⊂ R n. Here ∆p is the pLaplacia ..."
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Cited by 33 (3 self)
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The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane–Emden type, including the following two model problems: −∆pu = u q + µ, Fk[−u] = u q + µ, u ≥ 0, on R n, or on a bounded domain Ω ⊂ R n. Here ∆p is the pLaplacian defined by ∆pu = div (∇u∇u  p−2), and Fk[u] is the kHessian defined as the sum of k × k principal minors of the Hessian matrix D 2 u (k = 1, 2,...,n); µ is a nonnegative measurable function (or measure) on Ω. The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an open problem even for good data µ ∈ Ls (Ω), s> 1. Such results are deduced from our existence criteria with the sharp exponents s = n(q−p+1) pq for the first equation, and s = n(q−k) 2kq for the second one. Furthermore, a complete characterization of removable singularities is given. Our methods are based on systematic use of Wolff’s potentials, dyadic models, and nonlinear trace inequalities. We make use of recent advances in potential theory and PDE due to Kilpeläinen and Mal´y, Trudinger and Wang, and Labutin. This enables us to treat singular solutions, nonlocal operators, and distributed singularities, and develop the theory simultaneously for quasilinear equations and equations of MongeAmpère type.
The strong maximum principle revisited
 Journal of Differential Equations
"... Abstract. In this paper we first present the classical maximum principle due to E. Hopf [20], together with an extended commentary and discussion of Hopf’s paper (see also [37]). We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematica ..."
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Cited by 32 (10 self)
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Abstract. In this paper we first present the classical maximum principle due to E. Hopf [20], together with an extended commentary and discussion of Hopf’s paper (see also [37]). We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf’s principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here. In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results ([41], [30], [27]); and in extending the conclusions to wider classes of singular operators than previously considered. The results have unexpected ramifications for other problems, as will develop from the
Principal eigenvalues for some quasilinear elliptic equations onRn
 Adv. Differential Equations
, 1996
"... We improve some previous results for the principal eigenvalue of the plaplacian defined on IRN, study regularity of the corresponding eigenfunctions and give an existence result of the type below the first eigenvalue (or between the first eigenvalues) for a certain perturbed problem. Based in our a ..."
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Cited by 26 (4 self)
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We improve some previous results for the principal eigenvalue of the plaplacian defined on IRN, study regularity of the corresponding eigenfunctions and give an existence result of the type below the first eigenvalue (or between the first eigenvalues) for a certain perturbed problem. Based in our approach for the equation we deduce existence, ∗Key Phrases: pLaplacian systems, nonlinear eigenvalues problems, indefinite weight, homogenious Sobolev Spaces, unbounded domain, perturbation, maximum principle.
GRADIENT ESTIMATES VIA NONLINEAR POTENTIALS
, 906
"... Abstract. We present pointwise gradient bounds for solutions to pLaplacean type nonhomogeneous equations employing nonlinear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic equations. 1. Introduction and ..."
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Cited by 25 (3 self)
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Abstract. We present pointwise gradient bounds for solutions to pLaplacean type nonhomogeneous equations employing nonlinear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic equations. 1. Introduction and
VOLUME GROWTH, GREEN’S FUNCTIONS, AND PARABOLICITY OF ENDS
 VOL. 97, NO. 2 DUKE MATHEMATICAL JOURNAL
, 1999
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