Local Hölder regularity of gradients for evolutional <Emphasis Type="Italic">p</Emphasis>-Laplacian systems

We study a H?lder regularity of gradients for evolutional p-Laplacian systems with H?lder continuous coefficients and exterior force. We use the perturbation argument with the p-Laplacian systems with constant coefficients and only principal terms. The main task is to make the H?lder estimate of gradients for the systems above well-worked in the perturbation estimate. We also need to make a localization of the H?lder estimate in [2]. Received: June 25, 2001?Published online: July 9, 2002 Dedicated to Professor Norio Kikuchi on his sixtieth birthday This research was partially supported by the Grant-in-Aid for Encouragement of Young Scientists No. 12740102 at the Ministry of Educations, Science, Sports and Culture.     

Partial regularity for parabolic systems with non-standard growth

We study non-linear parabolic systems with non-standard p(z)-growth conditions and establish that the gradient of weak solutions is locally H?lder continuous with H?lder exponent b ? (0,1){\beta \in (0,1)} with respect to the parabolic metric on an open set of full Lebesgue measure, provided the exponent function p(z) itself is H?lder continuous with exponent β with respect to the parabolic metric.     

Hölder Regularity of Compactly Supported p -Wavelets: p =3,4,5

We provide exact asymptotic estimates for the H?lder regularity index of compactly supported minimal length p -wavelets in cases p=3,4,5 . February 28, 1996. Date revised: October 7, 1996.     

Persistence of bounded solutions of parabolic equations under domain perturbation

The aim of this paper is to study the behavior of bounded solutions of parabolic equations on the whole real line under perturbation of the underlying domain. We give the convergence of bounded solutions of linear parabolic equations in the L ² and the L ^p -settings. For the L ^p -theory, we also prove the H?lder regularity of bounded solutions with respect to time. In addition, we study the persistence of a class of bounded solutions which decay to zero at t → ±∞ of semilinear parabolic equations under domain perturbation.     

Fine regularity for elliptic systems with discontinuous ingredients

We propose results on interior Morrey, BMO and H?lder regularity for the strong solutions to linear elliptic systems of order 2b with discontinuous coefficients and right-hand sides belonging to the Morrey space L^p,λ. Received: 20 October 2004     

Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity q > 1 and with natural growth

In this paper we deal with the Hölder regularity up to the boundary of the solutions to a nonhomogeneous Dirichlet problem for second-order discontinuous elliptic systems with nonlinearity q > 1 and with natural growth. The aim of the paper is to clarify that the solutions of the above problem are always global Hölder continuous in the case of the dimension n = q without any kind of regularity assumptions on the coefficients. As a consequence of this sharp result, the singular sets $\Omega_0 \subset \OmegaIn this paper we deal with the H?lder regularity up to the boundary of the solutions to a nonhomogeneous Dirichlet problem for second-order discontinuous elliptic systems with nonlinearity q > 1 and with natural growth. The aim of the paper is to clarify that the solutions of the above problem are always global H?lder continuous in the case of the dimension n = q without any kind of regularity assumptions on the coefficients. As a consequence of this sharp result, the singular sets , are always empty for n = q. Moreover we show that also for 1 < q < 2, but q close enough to 2, the solutions are global H?lder continuous for n = 2.      

On the Hölder-Continuity of ω-minima

Summary We prove H?lder regularity for the gradient of local ω-minima of integral functionals with quadratic principal part.

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Riassunto In questo lavoro si dà un risultato di regolarità h?lderiana per il gradiente degli ω-minimi locali di funzionali integrali aventi parte principale quadratica.

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The author is member of G.N.A.F.A.; this work is supported by the Ministero della Pubblica Istruzione.     

Construction of Continuous Functions with Prescribed Local Regularity

In this paper we investigate from both a theoretical and a practical point of view the following problem: Let s be a function from [0;1] to [0;1] . Under which conditions does there exist a continuous function f from [0;1] to R such that the regularity of f at x , measured in terms of H?lder exponent, is exactly s(x) , for all x ∈ [0;1] ? We obtain a necessary and sufficient condition on s and give three constructions of the associated function f . We also examine some extensions regarding, for instance, the box or Tricot dimension or the multifractal spectrum. Finally, we present a result on the ``size' of the set of functions with prescribed local regularity. November 30, 1995. Date revised: September 30, 1996.     

Non variational basic elliptic systems of second order

Denoting byu a vector in R ^N defined on a bounded open set Ω ⊂ R ⁿ , we setH(u)={Dij u} and consider a basic differential operator of second ordera(H(u)) wherea(ξ) is a vector in R ^N , which is elliptic in the sense that it satisfies the condition (A). After a rapid comparison between this condition (A) and the classical definition of ellipticity, we shall prove that, if seu∈H ² (Ω) is a solution of the elliptic systema(H(u))=0 in Ω thenH(u)∈H _loc ^{2, q} for someq>2. We then deduce from this the so called fundamental internal estimates for the matrixH(u) and for the vectorsDu andu. We shall then present a first risult on h?lder regularity for the solutions of the system withf h?lder continuous in Ω, and a partial h?lder continuity risult for solutionsu∈H ² (Ω) of a differential systema (x, u, Du, H (u))=b(x, u, Du)     

Some new estimates for solutions of degenerate two dimensional Monge-Ampère equations

We derive a monotonicity formula for smooth solutions u of degenerate two dimensional Monge-Ampère equations, and use this to obtain a local H?lder gradient estimate, depending on for some . Received August 9, 1999; in final form December 8, 1999/ Published online December 8, 2000     

Partial regularity in non-linear elasticity

We prove partial regularity of minimizers of some polyconvex functionals. In particular our results include models such as ∫_Ω a(x,u)(|Du|²+| det Du|²), where a is a bounded H?lder continuous function, such that a(x,u)≥c for some positive constant c. Received: 2 January 2001 / Revised version: 30 August 2001     

Remarks on Hölder continuity for solutions of the p-Laplace evolution equations

We study the interior Hölder regularity problem for the gradient of solutions of the p-Laplace evolution equations with the external forces. Misawa gave some conditions for the Hölder continuity of the gradient of solutions. We show Hölder estimates of the solutions with weaker condition as for Misawa.     

On biharmonic maps and their generalizations

We give a new proof of regularity of biharmonic maps from four-dimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hölder inequalities is used next to prove continuity of solutions and higher integrability of their second order derivatives. As a byproduct, we also prove that a weak limit of biharmonic maps into a sphere is again biharmonic. The proof of regularity can be adapted to biharmonic maps on the Heisenberg group, and to other functionals leading to fourth order elliptic equations with critical nonlinearities in lower order derivatives.Received: 6 February 2003, Accepted: 12 March 2003, Published online: 16 May 2003Mathematics Subject Classification (2000): 35J60, 35H20Pawel Strzelecki: Current address (till September 2003): Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany (email: strzelec@math.uni-bonn.de). The author is partially supported by KBN grant no. 2-PO3A-028-22;he gratefully acknowledgesthe hospitality of his colleagues from Bonn,and the generosity of Humboldt Foundation.     

Onsager-Machlup functional for the fractional Brownian motion

 Let X be the solution of the stochastic differential equation where B ^H is a fractional Brownian motion with Hurst parameter H. In this paper we compute the Onsager-Machlup functional of X for the supremum norm and H?lder norms of order β with in the case and for H?lder norms of order β with when . Received: 16 July 2001 / Revised version: 12 March 2002 / Published online: 10 September 2002     

Regularity of quasi-minimizers on metric spaces

Using the theory of Sobolev spaces on a metric measure space we are able to apply calculus of variations and define p-harmonic functions as minimizers of the p-Dirichlet integral. More generally, we study regularity properties of quasi-minimizers of p-Dirichlet integrals in a metric measure space. Applying the De Giorgi method we show that quasi-minimizers, and in particular p-harmonic functions, satisfy Harnack's inequality, the strong maximum principle, and are locally H?lder continuous, if the space is doubling and supports a Poincaré inequality. Received: 12 May 2000 / Revised version: 20 April 2001     

The regularity for nonlinear parabolic systems of p-Laplacian type with critical growth

We study the Hölder regularity of weak solutions to the evolutionary p  -Laplacian system with critical growth on the gradient. We establish a natural criterion for proving that a small solution and its gradient are locally Hölder continuous almost everywhere. Actually our regularity result recovers the classical result in the case p=2$p=2$ [16] and can be applied to study the regularity of the heat flow for m-dimensional H-systems as well as the m-harmonic flow.     

Wavelets Techniques for Pointwise Anti-H?lderian Irregularity

In this paper, we introduce a notion of weak pointwise H?lder regularity, starting from the definition of the pointwise anti-H?lder irregularity. Using this concept, a weak spectrum of singularities can be defined as for the usual pointwise H?lder regularity. We build a class of wavelet series satisfying the multifractal formalism and thus show the optimality of the upper bound. We also show that the weak spectrum of singularities is disconnected from the casual one (referred to here as strong spectrum of singularities) by exhibiting a multifractal function made of Davenport series whose weak spectrum differs from the strong one.     

The G Class of Functions and Its Applications

Abstract In this paper, we introduce the concept of the G class of functions of the parabolic class, and show the H?lder continuity of the G class of functions. The introduction of this concept contributes to the proof of the regularity and existence of the solution for the first boundary problem of parabolic equation in divergence form. Project supported by NNFC (79790130) and ZJPNFC(198013) and STDF of Shanghai     

Partial regularity results for subelliptic systems in the Heisenberg group

We consider subelliptic systems in the Heisenberg group. We give a new proof for the smoothness of solutions of inhomogeneous systems with constant coefficients. With this result, we prove partial Hölder continuity of the horizontal gradient for non-linear systems with p-growth for p≥2 via the $\mathcal {A}We consider subelliptic systems in the Heisenberg group. We give a new proof for the smoothness of solutions of inhomogeneous systems with constant coefficients. With this result, we prove partial H?lder continuity of the horizontal gradient for non-linear systems with p-growth for p≥2 via the -harmonic approximation technique.     

Partial Hölder continuity for Q-valued energy minimizing maps

We consider multivalued maps between Ω ? ?^N open (N ≥ 2) and a smooth, compact Riemannian manifold 𝒩 locally minimizing the Dirichlet energy. An interior partial Hölder regularity results in the spirit of R. Schoen and K. Uhlenbeck is presented. Consequently a minimizer is Hölder continuous outside a set of Hausdorff dimension at most N ? 3. Almgren's original theory includes a global interior Hölder continuity result if the minimizers are valued into some ?^m. It cannot hold in general if the target is changed into a Riemannian manifold, since it already fails for “classical” single valued harmonic maps.