Convex bodies: the brunn–minkowski theory

On etudie la regularite limite des solutions faibles de l'equation divA(x,u,Du)+B(x,u,Du)=0 avec des conditions aux limites conormales et de Dirichlet

Boundary regularity for solutions of degenerate elliptic equations

In Section 21 the basic theory of sectorial operators is developed, including examples and the concept of sectorial approximation In Section 22 we introduce some notation for certain spaces of holomorphic functions on sectors A functional calculus for sectorial operators is constructed in Section 23 along the lines of the abstract framework of Chapter 1 Fundamental properties like the composition rule are proved In Section 25 we give natural extensions of the functional calculus to larger function spaces in the case where the given operator is bounded and/or invertible In this way a panorama of functional calculi is developed In Section 26 some mixed topics are discussed, eg, adjoints and restrictions of sectorial operators and some fundamental boundedness and some first approximation results Section 27 contains a spectral mapping theorem

The Functional Calculus for Sectorial Operators

완효성 비료와 4종복비의 혼용 시용이 팔레놉시스의 생육에 미치는 효과

The energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a 0-limit of three-dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v V U ! R n , U R n . We show that the L 2 -distance of rv from a single rotation matrix is bounded by a multiple of the L 2 -distance from

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A Theorem on Geometric Rigidity and the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity

On montre que pour p fixe, 1<p<∞, si n est une solution faible de ⊇•(|⊇u| P−2 ⊇u)=0 dans un domaine borne D⊆R n alors ⊇u est continue de Holder sur des sous-ensembles compacts de D

Regularity of the derivatives of solutions to certain degenerate elliptic equations

We show that the gradient of a solution to a parabolic system of p-Laplacian type in Rn satisfies a reverse Hölder inequality provided p > 2n/(n+2). In particular, this implies the local higher integrability of the gradient.

Higher integrability for parabolic systems of p-Laplacian type

In this paper we study closed sets E which are "locally uniformly fat" with respect to a certain nonlinear Riesz capacity. We show that E is actually "locally uniformly fat" with respect to a weaker Riesz capacity. Two applications of this result are given. The first application is concerned with proving Sobolev-type inequalities in domains whose complements are uniformly fat. The second application is concerned with the Fekete points of E. Introduction. Let x = (X ,X2, ... , x) be a point in Euclidean n space, Rn, with jxl the norm of x. For a > 0, let I,>(x) = IXI(a-n) denote the Riesz kernel of order a and put I, * f(x) = f Ix _ y(&f-n)f(y) dy, x E R , Rn IC, * ,u(x) = fR _ -yj(c-n) d,u(y), x E R , Rn when f is a Lebesgue integrable function and ,u a Borel measure on Rn. Here dy denotes Lebesgue measure on Rn. If E C Rn, 0 1, define the (a, p) outer Riesz capacity of E by Ra,P(E) = inf{jfllfP: Ia, * f > 1 on El, where If lIp is the Lebesgue p norm of f. We note that a = 1, p = 2, is the classical Newtonian capacity. If ap = n, we let Rc,p(E) = inf{jfjjf|: J, * f > 1 on El, where Jc, is the truncated Riesz kernel defined by Je(x) = IXI(a-n) _ (100)(a-n), |x ? 100, = 0, lxl > 100. We shall say that a property holds (a, p) quasi everywhere (abbreviated (a, p) q.e.) on a set E if it holds on E except perhaps for a set of Rop capacity zero. Next we list some properties of Riesz capacities. To simplify the writing we state these properties only for 0 * u)l/(P-l) > 1, (a,p) q.e., on E and P = 1, (a,p) q.e., on supp,u C E. Moreover, _ _(E) = jjlI * pI,ujj = Rop(E) Received by the editors March 11, 1987. 1980 Mathematic3 Subject Cla33ification (1985 Revi3ion). Primary 31C15, 31B15. (?1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

Uniformly fat sets

Capacitary functions in convex rings

(1993). On very weak solutions of certain elliptic systems. Communications in Partial Differential Equations: Vol. 18, No. 9-10, pp. 1515-1537.

John L. Lewis

Papers

Regularity of the derivatives of solutions to certain degenerate elliptic equations

Higher integrability for parabolic systems of p-Laplacian type

Uniformly fat sets

Capacitary functions in convex rings

On very weak solutions of certain elliptic systems