A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing in the class of functions W1,G(Ω) with , for a given φ₀?0 and bounded. W1,G(Ω) is the class of weakly differentiable functions with . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1,α regularity of their free boundaries near “flat” free boundary points.     

Two results on entire solutions of Ginzburg-Landau system in higher dimensions

In this short article we prove two results on the Ginzburg-Landau system of equations Δu=u(|u|²−1), where . First we prove a Liouville-type theorem which asserts that every solution u, satisfying , is constant (and of unit norm), provided N?4 (here M?1). In our second result, we give an answer to a question raised by Brézis (open problem 3 of (Proceedings of the Symposium on Pure Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1999), about the symmetry for the Ginzburg-Landau system in the case N=M?3. We also formulate three open problems concerning the classification of entire solutions of the Ginzburg-Landau system in any dimension.     

Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary

We present some new regularity criteria for “suitable weak solutions” of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are Hölder continuous up to the boundary provided that the scaled mixed norm with 3/p+2/q?2, 2<q?∞, (p,q)≠(3/2,∞) is small near the boundary. Our methods yield new results in the interior case as well. Partial regularity of weak solutions is also analyzed under some additional integral conditions.     

Lipschitz and piecewise-C regularity for scalar minimizers of affine simple integrals

Lipschitz, piecewise-C¹ and piecewise affine regularity is proved for AC minimizers of the “affine” integral , under general hypotheses on , , and with superlinear growth at infinity.The hypotheses assumed to obtain Lipschitz continuity of minimizers are unusual: ρ(·) and ?(·) are lsc and may be both locally unbounded (e.g., not in L_loc¹), provided their quotient ?/ρ(·) is locally bounded. As to h(·), it is assumed lsc and may take +∞ values freely.     

Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem

We deal with the Cauchy problem for a strictly hyperbolic second-order operator with non-regular coefficients in the time variable. It is well-known that the problem is well-posed in L² in case of Lipschitz continuous coefficients and that the log-Lipschitz continuity is the natural threshold for the well-posedness in Sobolev spaces which, in this case, holds with a loss of derivatives. Here, we prove that any intermediate modulus of continuity between the Lipschitz and the log-Lipschitz one leads to an energy estimate with arbitrary small loss of derivatives. We also provide counterexamples to show that the following classification:

     

Planar nonautonomous polynomial equations II. Coinciding sectors

We give a few sufficient conditions for the existence of periodic solutions of the equation where aj's are complex-valued. We prove the existence of one up to two periodic solutions and heteroclinic ones.     

Approximation numbers for the finite sections of Toeplitz operators with piecewise continuous symbols

In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator T(a)∈L(?p), 1<p<∞, where a is a piecewise continuous function on the unit circle. We prove that the behavior of the approximation numbers of the finite sections Tn(a)=PnT(a)Pn depends heavily on the Fredholm properties of the operators T(a) and . In particular, if the operators T(a) and are Fredholm on ?p, then the approximation numbers of Tn(a) have the so-called k-splitting property. But, in contrast with the case of continuous symbols, the splitting number k is in general larger than .     

Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations

We consider the Dirichlet problem for positive solutions of the equation −Δ_m(u)=f(u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak solutions. In particular when f(s)>0 for s>0 we prove summability properties of , and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |Du|^m−2. The point of view of considering |Du|^m−2 as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov-Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f(s)>0 for s>0 and m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<m<2.     

Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations

We present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three: a suitable weak solution is regular near an interior point z if the scaled -norm of the velocity with 1?3/p+2/q?2, 1?q?∞ is sufficiently small near z and if the scaled -norm of the magnetic field with 1?3/l+2/m?2, 1?m?∞ is bounded near z. Similar results are also obtained for the vorticity and for the gradient of the vorticity. Furthermore, with the aid of the regularity criteria, we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations.     

Analysis of sharp polynomial upper estimate of number of positive integral points in a five-dimensional tetrahedra

Let a?b?c?d?e?1 be real numbers and P₅ be the number of positive integral solutions of . In this paper we show that 120P₅?(a-1)(b-1)(c-1)(d-1)(e-1). This confirms a conjecture of Durfee for the dimension 5 case. We show also that the upper estimate of P₅ given by Lin and Yau is strictly sharper than that suggested by Durfee conjecture if , but is not sharper than that suggested by Durfee conjecture if .     

Viscosity solutions for partial differential equations with Neumann type boundary conditions and some aspects of Aubry-Mather theory

We study partial differential inequalities (PDI) of the type where NK(⋅) is the normal cone to the set K. We prove existence of a constant such that the PDI of Hamilton-Jacobi type has a unique (global) Lipschitz viscosity solution. We provide a formula to calculate this constant. Moreover, we define a subset of K such that any two solutions of the previous PDI which coincide on will coincide on K. Our paper generalizes results of the case without boundary conditions for convex Hamiltonians obtained by L.C. Evans and A. Fathi.     

Barycentric selectors and a Steiner-type point of a convex body in a Banach space

Let F be a mapping from a metric space into the family of all m-dimensional affine subsets of a Banach space X. We present a Helly-type criterion for the existence of a Lipschitz selection f of the set-valued mapping F, i.e., a Lipschitz continuous mapping satisfying . The proof of the main result is based on an inductive geometrical construction which reduces the problem to the existence of a Lipschitz (with respect to the Hausdorff distance) selector S_X^(m) defined on the family of all convex compacts in X of dimension at most m. If X is a Hilbert space, then the classical Steiner point of a convex body provides such a selector, but in the non-Hilbert case there is no known way of constructing such a point. We prove the existence of a Lipschitz continuous selector for an arbitrary Banach space X. The proof is based on a new result about Lipschitz properties of the center of mass of a convex set.     

Regularity theorems for parabolic equations

We study regularity properties of solutions of a parabolic equation in R₊×R^d, d?3 under fairly general conditions on the drift term coefficients. The results are already new for the case a=I, , b=b(x) and .     

Integral of the Clarke subdifferential mapping and a generalized Newton-Leibniz formula

In the theory of Lebesgue integration it has been proved that if f is a real Lipschitz function defined on a segment [a,b]⊂R, then the Newton-Leibniz formula (the fundamental theorem of calculus) holds. This paper extends the fact to the case where the Fréchet derivative f^′(⋅) (which is defined almost everywhere on [a,b] by the Rademacher theorem) and the Lebesgue integral are replaced, respectively, by the Clarke subdifferential mapping ∂_Cf(⋅) and the Aumann (set-valued) integral. Among other things, we show that and the equality is valid if and only if f is strictly Hadamard differentiable almost everywhere on [a,b]. The result is derived from a general representation formula, which we obtain herein for the integral of the Clarke subdifferential mapping of a Lipschitz function defined on a separable Banach space.     

Optimal uniqueness theorems and exact blow-up rates of large solutions

In this paper, we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non-linearity and combine these results with the localization method introduced in [López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25-45] to prove that any large solution L of Δu=a(x)u^p, p>1, a>0, must satisfy

     

Dynamics of the viscous Cahn-Hilliard equation

We study generalized viscous Cahn-Hilliard problems with nonlinearities satisfying critical growth conditions in , where Ω is a bounded smooth domain in Rn, n?3. In the critical growth case, we prove that the problems are locally well posed and obtain a bootstrapping procedure showing that the solutions are classical. For p=2 and almost critical dissipative nonlinearities we prove global well posedness, existence of global attractors in and, uniformly with respect to the viscosity parameter, L^∞(Ω) bounds for the attractors. Finally, we obtain a result on continuity of regular attractors which shows that, if n=3,4, the attractor of the Cahn-Hilliard problem coincides (in a sense to be specified) with the attractor for the corresponding semilinear heat equation.     

Classification of Razor Blades to the filtration equation—the sublinear case

We study nonnegative solutions of the filtration equation u_t=Δ?(u) in , where ? is continuous, increasing and sublinear. More precisely, we study the Razor Blades, i.e., solutions which may be singular at |x|=0 for t>0, and start with zero initial data. We first prove a nonexistence result when ? is too sublinear and we show an axial trace result in the other case: there exist a time τ≡τ(u) and a Radon measure λ on [0,τ) such that