Regularity of radial minimizers of reaction equations involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian

We consider semi-stable, radially symmetric, and decreasing solutions of  − Δ _p u = g(u) in the unit ball of , where p > 1, Δ _p is the p-Laplace operator, and g is a locally Lipschitz function. For this class of radial solutions, which includes local minimizers, we establish pointwise, L ^q , and W ^1,q estimates which are optimal and do not depend on the specific nonlinearity g. Among other results, we prove that every radially decreasing and semi-stable solution u belonging to W ^1,p (B ₁) is bounded whenever n < p + 4p/(p − 1). Under standard assumptions on the nonlinearity g(u) = λf (u), where λ > 0 is a parameter, it is proved that the corresponding extremal solution u ^* is semi-stable, and hence, it enjoys the regularity stated in our main result.     

Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions

Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂_ν u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L ^∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s| ^p ) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) =  s ^p then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of ∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential equation is of the form h(x, u) with h satisfying suitable growth conditions.     

Fast and slow decay solutions for supercritical elliptic problems in exterior domains

We consider the elliptic problem Δu  +  u ^p  =  0, u  >  0 in an exterior domain, under zero Dirichlet and vanishing conditions, where is smooth and bounded in , N ≥ 3, and p is supercritical, namely . We prove that this problem has infinitely many solutions with slow decay at infinity. In addition, a solution with fast decay O(|x|^2-N ) exists if p is close enough from above to the critical exponent.     

Elliptic Problems with Singular Potential and Double-Power Nonlinearity

We prove the existence of positive symmetric solutions to the semilinear elliptic problem

On scalar field equations with critical nonlinearity

The paper concerns existence of solutions to the scalar field equation

Multilevel frames for sparse tensor product spaces

For Au = f with an elliptic differential operator and stochastic data f, the m-point correlation function of the random solution u satisfies a deterministic equation with the m-fold tensor product operator A ^(m) of A. Sparse tensor products of hierarchic FE-spaces in are known to allow for approximations to which converge at essentially the rate as in the case m = 1, i.e. for the deterministic problem. They can be realized by wavelet-type FE bases (von Petersdorff and Schwab in Appl Math 51(2):145–180, 2006; Schwab and Todor in Computing 71:43–63, 2003). If wavelet bases are not available, we show here how to achieve the fast computation of sparse approximations of for Galerkin discretizations of A by multilevel frames such as BPX or other multilevel preconditioners of any standard FEM approximation for A. Numerical examples illustrate feasibility and scope of the method.     

Poincaré inequality and Palais–Smale condition for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

An improved Poincaré inequality and validity of the Palais-Smale condition are investigated for the energy functional on , 1 < p < ∞, where Ω is a bounded domain in , is a spectral (control) parameter, and is a given function, in Ω. Analysis is focused on the case λ = λ₁, where −λ₁ is the first eigenvalue of the Dirichlet p-Laplacian Δ _p on , λ₁ > 0, and on the “quadratization” of within an arbitrarily small cone in around the axis spanned by , where stands for the first eigenfunction of Δ _p associated with −λ₁.     

Multiplicity of Positive Solutions for a Class of Inhomogeneous Neumann Problems Involving the p(x)-Laplacian

We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving the p(x)-Laplacian of the form

Single-point condensation phenomena for a four-dimensional biharmonic Ren–Wei problem

We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E _p ): as p gets large, where Ω is a smooth bounded domain in R ⁴ . In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc. AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky pattern, under the assumption that the domain is convex.     

A weighted eigenvalue problem for the p-Laplacian plus a potential

Let Δ _p denote the p-Laplacian operator and Ω be a bounded domain in . We consider the eigenvalue problem

Large solutions to the p-Laplacian for large p

In this work we consider the behaviour for large values of p of the unique positive weak solution u _p to Δ _p u = u ^q in Ω, u = +∞ on , where q > p − 1. We take q = q(p) and analyze the limit of u _p as p → ∞. We find that when q(p)/p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of with u = +∞ on . If Q = 1 then solutions go to ∞ in the whole Ω and when Q = ∞ solutions converge to 1 uniformly in compact subsets of Ω, hence the boundary blow-up is lost in the limit.     

Some results on the qualitative properties of positive solutions of quasilinear elliptic equations

We consider the Dirichlet problem in Ω with zero Dirichlet boundary conditions. We prove local summability properties of and we exploit these results to give geometric characterizations of the critical set . We extend to the case of changing sign nonlinearities some results known in the case f(s) > 0 for s > 0. Berardino Sciunzi: Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”     

Optimal initial value conditions for the existence of local strong solutions of the Navier–Stokes equations

Consider the instationary Navier–Stokes system in a smooth bounded domain with vanishing force and initial value . Since the work of Kiselev and Ladyzhenskaya (Am. Math. Soc. Transl. Ser. 2 24:79–106, 1963) there have been found several conditions on u ₀ to prove the existence of a unique strong solution with u(0) = u ₀ in some time interval [0, T), 0 < T ≤ ∞, where the exponents 2 < s < ∞, 3 < q < ∞ satisfy . Indeed, such conditions could be weakened step by step, thus enlarging the corresponding solution classes. Our aim is to prove the following optimal result with the weakest possible initial value condition and the largest possible solution class: Given u ₀, q, s as above and the Stokes operator A ₂, we prove that the condition is necessary and sufficient for the existence of such a local strong solution u. The proof rests on arguments from the recently developed theory of very weak solutions.     

On the Multiplicity of Nodal Solutions to a Singularly Perturbed Neumann Problem

We consider the problem

On rectifiable curves with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bounds on global curvature: self-avoidance,regularity, and minimizing knots

We discuss the analytic properties of curves γ whose global curvature function ρ _G [γ]⁻¹ is p-integrable. It turns out that the L ^p -norm is an appropriate model for a self-avoidance energy interpolating between “soft” knot energies in form of singular repulsive potentials and “hard” self-obstacles, such as a lower bound on the global radius of curvature introduced by Gonzalez and Maddocks. We show in particular that for all p > 1 finite -energy is necessary and sufficient for W ^2,p -regularity and embeddedness of the curve. Moreover, compactness and lower-semicontinuity theorems lead to the existence of -minimizing curves in given isotopy classes. There are obvious extensions to other variational problems for curves and nonlinearly elastic rods, where one can introduce a bound on to preclude self-intersections.     

Morse indices and the number of maximum points of some solutions to a two-dimensional elliptic problem

In this note, we consider the problem

Anti-selfdual Lagrangians II: unbounded non self-adjoint operators and evolution equations

New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form as well ass dissipative initial value evolutions of the form where is a convex potential on an infinite dimensional space, A is a linear operator and is any scalar. The framework developed in the above mentioned paper reformulates these problems as and respectively, where is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term. Nassif Ghoussoub research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The author gratefully acknowledges the hospitality and support of the Centre de Recherches Mathématiques in Montréal where this work was initiated. Leo Tzou’s research was partially supported by a doctoral postgraduate scholarship from the Natural Science and Engineering Research Council of Canada.     

Estimates of holomorphic functions in zero-free domains

We study functions f(z) holomorphic in having the property f(z) ≠ 0 for 0 < Im z < 1 and we obtain lower bounds for |f(z)| for 0 < Im z < 1. In our analysis we deal with scalar functions f(z) as well as with operator valued holomorphic functions I + A(z) assuming that A(z) is a trace class operator for and I + A(z) is invertible for 0 < Im z < 1 and is unitary for . A. Borichev was partially supported by the ANR project DYNOP.     

Gradient estimates for the Perona–Malik equation

We consider the Cauchy problem for the Perona–Malik equation