Bernhard Ruf - Academia.edu (original) (raw)

Papers by Bernhard Ruf

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Advances in Nonlinear Analysis, 2014

In this article we present recent results on optimal embeddings, and associated PDEs, of the spac... more In this article we present recent results on optimal embeddings, and associated PDEs, of the space of functions whose distributional Laplacian belongs to L1 . We discuss sharp embedding inequalities which allow to improve the optimal summability results for solutions of Poisson equations with L1 -data by Maz'ya (N ≥ 3) and Brezis–Merle (N = 2). Then, we consider optimal embeddings of the mentioned space into L1 , for the simply supported and the clamped case, which yield corresponding eigenvalue problems for the 1-biharmonic operator (a higher order analogue of the 1-Laplacian). We derive some properties of the corresponding eigenfunctions, and prove some Faber–Krahn type inequalities.

Evolution Equations, Semigroups and Functional Analysis, 2002

Progress in Nonlinear Differential Equations and Their Applications, 2014

The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

ZAMP Journal of Applied Mathematics and Physics, 1989

Transactions of the American Mathematical Society, 2012

Adams’ inequality for bounded domains Ω ⊂ R 4 \Omega \subset \mathbb {R}^4 states that the suprem... more Adams’ inequality for bounded domains Ω ⊂ R 4 \Omega \subset \mathbb {R}^4 states that the supremum of ∫ Ω e 32 π 2 u 2 d x \int _{\Omega } e^{32 \pi ^2 u^2} \, dx over all functions u ∈ W 0 2 , 2 ( Ω ) u \in W_0^{2, \, 2}(\Omega ) with ‖ Δ u ‖ 2 ≤ 1 \| \Delta u\|_2 \leq 1 is bounded by a constant depending on Ω \Omega only. This bound becomes infinite for unbounded domains and in particular for R 4 \mathbb {R}^4 . We prove that if ‖ Δ u ‖ 2 \|\Delta u\|_2 is replaced by a suitable norm, namely ‖ u ‖ := ‖ − Δ u + u ‖ 2 \| u \|:=\|- \Delta u + u\|_2 , then the supremum of ∫ Ω ( e 32 π 2 u 2 − 1 ) d x \int _{\Omega } (e^{32 \pi ^2 u^2} -1) \, dx over all functions u ∈ W 0 2 , 2 ( Ω ) u \in W_0^{2, \, 2}(\Omega ) with ‖ u ‖ ≤ 1 \|u\| \leq 1 is bounded by a constant independent of the domain Ω \Omega . Furthermore, we generalize this result to any W 0 m , n m ( Ω ) W_0^{m, \, \frac n m}(\Omega ) with Ω ⊆ R n \Omega \subseteq \mathbb {R}^{n} and m m an even integer less than n n .

Proceedings of the Steklov Institute of Mathematics, 2006

In this paper we consider nonlinear elliptic equations of the form −∆u = g(u) , in Ω, u = 0 , on ... more In this paper we consider nonlinear elliptic equations of the form −∆u = g(u) , in Ω, u = 0 , on ∂Ω , and Hamiltonian type systems of the form :        −∆u = g(v) , in Ω, −∆v = f (u) , in Ω, u = 0 and v = 0 , on ∂Ω, where Ω is a bounded domain in R 2 , and f , g ∈ C(R) are superlinear nonlinearities. In two dimensions the maximal growth (= critical growth) of f , g (such that the problem can be treated variationally) is of exponential type, given by Pohozaev-Trudinger type inequalities. We discuss existence and nonexistence results related to critical growth for the equation and the system. The natural framework for such equations and systems are Sobolev spaces, which give in most cases an adequate answer concerning the maximal growth involved. However, we will see that for the system in dimension 2, the Sobolev embeddings are not sufficiently fine to capture the true maximal growths. We will show that working in Lorentz spaces gives better results.

Nonlinear Analysis: Theory, Methods & Applications, 2012

Multiplicity results are proved for the nonlinear elliptic system    −∆u + g(v) = 0 −∆v + g(u)... more Multiplicity results are proved for the nonlinear elliptic system    −∆u + g(v) = 0 −∆v + g(u) = 0 in Ω, u = v = 0 on ∂Ω, (1) where Ω ⊂ R N is a bounded domain with smooth boundary and g : R −→ R is a nonlinear C 1-function which satisfies addtional conditions. No assumption of symmetry on g is imposed. Extensive use is made of a global version of the Lyapunov-Schmidt reduction method due to Castro and Lazer (see [C] and [CL]), and of symmetric versions of the Mountain Pass Theorem (see [AR] and [R]).

Mediterranean Journal of Mathematics, 2004

In this paper we study the existence of nontrivial solutions for the following system of coupled ... more In this paper we study the existence of nontrivial solutions for the following system of coupled semilinear Poisson equations:        −∆u = v p , in Ω, −∆v = f (u) , in Ω, u = 0 and v = 0 , on ∂Ω, where Ω is a bounded domain in R N. We assume that 0 < p < 2 N −2 , and the function f is superlinear and with no growth restriction (for example f (s) = s e s); then the system has a nontrivial (strong) solution. Vol. 99 (9999) Elliptic Systems 3

We consider elliptic equations in bounded domains RN with non- linearities which have critical gr... more We consider elliptic equations in bounded domains RN with non- linearities which have critical growth at +1 and linear growth at1 , with > 1, the rst eigenvalue of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms provided N 6. In dimensions N = 3; 4; 5 an additional lower order growth term has to be added to the nonlinearity, similarly as in the famous result of Brezis-Nirenberg for equations with critical growth.

Journal of Differential Equations, 2007

The Fučík spectrum for systems of second order ordinary differential equations with Dirichlet or ... more The Fučík spectrum for systems of second order ordinary differential equations with Dirichlet or Neumann boundary values is considered: it is proved that the Fučík spectrum consists of global C 1 surfaces, and that through each eigenvalue of the linear system pass exactly two of these surfaces. Further qualitative, asymptotic and symmetry properties of these spectral surfaces are given. Finally, related problems with nonlinearities which cross asymptotically some eigenvalues, as well as linear-superlinear systems are studied.

Communications in Contemporary Mathematics, 2010

In this note we consider the eigenvalue problem for the Laplacian with the Neumann and Robin boun... more In this note we consider the eigenvalue problem for the Laplacian with the Neumann and Robin boundary conditions involving the Hardy potential. We prove the existence of eigenfunctions of the second eigenvalue for the Neumann problem and of the principal eigenvalue for the Robin problem in "high" dimensions.

Calculus of Variations and Partial Differential Equations, 1995

Calculus of Variations and Partial Differential Equations, 2011

Let Ω be a bounded, smooth domain in R 2. We consider the functional I(u) = Ω e u 2 dx in the sup... more Let Ω be a bounded, smooth domain in R 2. We consider the functional I(u) = Ω e u 2 dx in the supercritical Trudinger-Moser regime, i.e. for Ω |∇u| 2 dx > 4π. More precisely, we are looking for critical points of I(u) in the class of functions u ∈ H 1 0 (Ω) such that Ω |∇u| 2 dx = 4 π k (1+α), for small α > 0. In particular, we prove the existence of 1-peak critical points of I(u) with Ω |∇u| 2 dx = 4π(1 + α) for any bounded domain Ω, 2-peak critical points with Ω |∇u| 2 dx = 8π(1 + α) for non-simply connected domains Ω, and k-peak critical points with Ω |∇u| 2 dx = 4kπ(1 + α) if Ω is an annulus.

Journal of Functional Analysis, 2010

Let Ω be a bounded, smooth domain in R 2. We consider critical points of the Trudinger-Moser type... more Let Ω be a bounded, smooth domain in R 2. We consider critical points of the Trudinger-Moser type functional J λ (u) = 1 2 Ω |∇u| 2 − λ 2 Ω e u 2 in H 1 0 (Ω), namely solutions of the boundary value problem u + λue u 2 = 0 with homogeneous Dirichlet boundary conditions, where λ > 0 is a small parameter. Given k 1 we find conditions under which there exists a solution u λ which blows up at exactly k points in Ω as λ → 0 and J λ (u λ) → 2kπ. We find that at least one such solution always exists if k = 2 and Ω is not simply connected. If Ω has d 1 holes, in addition d + 1 bubbling solutions with k = 1 exist. These results are existence counterparts of one by Druet in [O. Druet, Multibump analysis in dimension 2: Quantification of blow-up levels, Duke Math. J. 132 (2) (2006) 217-269] which classifies asymptotic bounded energy levels of blow-up solutions for a class of nonlinearities of critical exponential growth, including this one as a prototype case.

Indiana University Mathematics Journal, 2003

The main purpose of this paper is to establish the existence of a solution of the semilinear Schr... more The main purpose of this paper is to establish the existence of a solution of the semilinear Schrödinger equation −∆u + V (x)u = f (u), in R 2 where V is a 1-periodic functions with respect to x, 0 lies in a gap of the spectrum of −∆ + V , and f (s) behaves like ± exp(αs 2) when s → ±∞.

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2014

We consider the problem of finding the optimal constant for the embedding of the space W 2,1 1 ()... more We consider the problem of finding the optimal constant for the embedding of the space W 2,1 1 () := n u 2 W 1,1 0 () | 1u 2 L 1 () o into the space L 1 (), where ✓ R n is a bounded convex domain, or a bounded domain with boundary of class C 1,↵. This is equivalent to finding the first eigenvalue of the 1-biharmonic operator under (generalized) Navier boundary conditions. In this paper we provide an interpretation for the eigenvalue problem, we show some properties of the first eigenfunction, we prove an inequality of Faber-Krahn type, and we compute the first eigenvalue and the associated eigenfunction explicitly for a ball, and in terms of the torsion function for general domains.

Matemática Contemporânea, 2008

We first survey some recent results on optimal embeddings for the space of functions with ∆u ∈ L ... more We first survey some recent results on optimal embeddings for the space of functions with ∆u ∈ L 1 (Ω), where Ω ⊂ R 2 is a bounded domain. The target space in the embeddings turns out to be a Zygmund space and the best constants are explicitly known. Remarkably, the best constant in the case of zero boundary data is twice the best constant in the case of compactly supported functions. Then, following the same strategy, we establish a new version of the celebrated Trudinger-Moser inequality, as embedding into the Zygmund space Z 1/2 0 (Ω), and we prove that, in contrast to the Moser case, here the best embedding constant is not attained.

In this paper we prove that the equation du dt + n i=0 ai(t)u i = f (t), t ∈ [0, 1], u(0) = u(1),... more In this paper we prove that the equation du dt + n i=0 ai(t)u i = f (t), t ∈ [0, 1], u(0) = u(1), has for every continuous f at most n solutions provided that n is odd, and the continuous coefficients ai satisfy |an(t)| ≥ α > 0 and |ai(t)| ≤ β, i = 1,. .. , n − 1, with β > 0 sufficiently small. Furthermore, we show that this result implies that for a restricted subclass of polynomial vector fields of order n in R 2 the maximal number of limit cycles is n. This constitutes a special case of Hilbert's 16th problem.

l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute ut... more l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/