ISSN:

1534-0392

eISSN:

1553-5258

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## Communications on Pure & Applied Analysis

2010 , Volume 9 , Issue 5

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*+*[Abstract](213)

*+*[PDF](18.8KB)

**Abstract:**

The 6th european conference on elliptic and parabolic problems took place in Gaeta from May 25 to May 29, 2009. It brought together more than 170 participants. This volume collects some of the papers presented there.

This meeting could not have been possible without the support of the Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, the Università di Cassino, the Accademia Pontaniana (Napoli), the Istituto Italiano per gli Studi Filosoci (Napoli), the GNAMPA, the Université de Haute Alsace (Mulhouse), the Universität Zürich, the MeMoMat Sapienza Università di Roma, the IAC CNR, the Comune di Gaeta and the partial support of the ERC grant 207573-2 Vectorial Problems. We thank all these institutions for their help.

We would like also to thank DCDS and especially Professor Shouchuan Hu for having accepted to publish these articles.

*+*[Abstract](326)

*+*[PDF](1162.2KB)

**Abstract:**

We study the large time behavior of the weak solutions to a one-phase moving sharp-interface PDE system describing the aggressive penetration of gaseous carbon dioxide in unsaturated concrete. The key of the proof is a global uniform estimate for solutions obtained by using the maximum principle. The analysis reported here relies on the global existence and uniqueness of solutions that we have proved previously.

*+*[Abstract](294)

*+*[PDF](443.3KB)

**Abstract:**

We derive a macroscopic model of electrical conduction in biological tissues in the high radio-frequency range, which is relevant in applications like electric impedance tomography. This model is derived via a homogenization limit by a microscopic formulation, based on Maxwell’s equations, taking into account the periodic geometry of the microstructure. We also study the asymptotic behavior of the solution for large times. Our results imply that periodic boundary data lead to an asymptotically periodic solution.

*+*[Abstract](381)

*+*[PDF](392.4KB)

**Abstract:**

In this paper we treat the question of the non--existence of global solutions, or their long time behavior, of nonlinear hyperbolic Kirchhoff systems. The main $p$--Kirchhoff operator may be affected by a perturbation which behaves like $|u|^{p-2} u$ and the systems also involve an external force $f$ and a nonlinear boundary damping $Q$. When $p=2$, we consider some problems involving a higher order dissipation term, under dynamic boundary conditions. For them we give criteria in order that $ || u(t,\cdot) ||_q\to\infty$ as $t \to\infty$ along any global solution $u=u(t,x)$, where $q$ is a parameter related to the growth of $f$ in $u$. Special subcases of $f$ and $Q$, interesting in applications, are presented in Sections 4, 5 and 6.

*+*[Abstract](239)

*+*[PDF](987.0KB)

**Abstract:**

We consider elliptic boundary value problems on large spherical caps with parameter dependent power nonlinearities. In this paper we show that imperfect bifurcation occurs as in the work [13]. When the domain is the whole sphere, there is a constant solution. In the case where the domain is a spherical cap, however, the constant solution disappears due to the boundary condition. For large spherical caps we construct solutions which are close to the constant solution in the whole n-dimensional sphere, using the eigenvalues of the linearized problem in the whole sphere and fixed point arguments based on a Lyapunov-Schmidt type reduction. Numerically there is a surprising similarity between the diagrams of this problem and the ones obtained in [18], also [5], for a Brezis-Nirenberg type problem on spherical caps.

*+*[Abstract](257)

*+*[PDF](188.4KB)

**Abstract:**

We study a one-phase Stefan problem for a semi-infinite material with temperature-dependent thermal conductivity and a convective term with a convective boundary condition at the fixed face $x=0$. We obtain sufficient conditions for data in order to have a parametric representation of the solution of the similarity type for $t \geq t_0 > 0$ with $t_0 $ an arbitrary positive time. We obtain explicit solutions through the unique solution of a Cauchy problem with the time as a parameter and we also give an algorithm in order to compute the explicit solution.

*+*[Abstract](252)

*+*[PDF](278.9KB)

**Abstract:**

We consider mixed Dirichlet-Robin problems on scale irregular domains. In particular, we study the asymptotic convergence of the solutions of elliptic problems with Robin boundary conditions on the "prefractal" curves approximating the scale irregular fractals.

*+*[Abstract](233)

*+*[PDF](215.0KB)

**Abstract:**

The model of a rigid linear heat conductor with memory is analyzed. Specifically, an evolution problem which describes the time evolution of the temperature distribution within a rigid heat conductor with memory is studied. The attention is focussed on the thermodynamical state of such a rigid heat conductor which, according to the adopted constitutive equations, depends on the history of the material; indeed, the dependence of the heat flux on the history of the temperature’s gradient is modeled via an integral term. Thus, the evolution problem under investigation is an integro-differential one with assigned initial and boundary conditions. Crucial in the present study are suitable expressions of an appropriate free energy and thermal work, related one to the other, which allow to construct functional spaces which are meaningful both under the physical as well as the analytic viewpoint. On the basis of existence and uniqueness results previously obtained, exponential decay at infinity is proved.

*+*[Abstract](309)

*+*[PDF](170.6KB)

**Abstract:**

The paper is concerned with the existence of solutions to an integrodifferential problem arising in the neutron transport theory. By an anisotropic singular perturbations method we show that solutions of such a problem exist and are close to those of some nonlocal elliptic problem. The existence of the solutions of the nonlocal elliptic problem with bounded data is ensured by the Schauder fixed point theorem. Then an asymptotic method is applied in the general case. The limits of the solutions of the nonlocal elliptic problems are solutions of our integro-differential problem.

*+*[Abstract](250)

*+*[PDF](263.6KB)

**Abstract:**

We consider the magnetic NLS equation

$ (-\varepsilon i \nabla+A(x)) ^2 u+V(x)u=K(x) |u|^{p-2}u, \quad x\in R^N, $

where $N \geq 3$, $2 < p < 2^*: = 2N/(N-2)$, $A:R^N\to R^N$ is a magnetic potential and $V: R^N \to R$, $K:R^N \to R$ are bounded positive potentials. We consider a group $G$ of orthogonal transformations of $ R^N$ and we assume that $A$ is $G$-equivariant and $V$, $K$ are $G$-invariant. Given a group homomorphism $\tau:G\to S^1$ into the unit complex numbers we look for semiclassical solutions $u_{\varepsilon}: R^N\to C$ to the above equation which satisfy

$ u_{\varepsilon}(gx)=\tau(g)u_{\varepsilon}(x)$

for all $g\in G$, $x\in R^N$. Using equivariant Morse theory we obtain a lower bound for the number of solutions of this type.

*+*[Abstract](194)

*+*[PDF](190.3KB)

**Abstract:**

We provide an informal overview on the theory of transport equa- tions with non smooth velocity fields, and on some applications of this theory to the well-posedness of hyperbolic systems of conservation laws.

*+*[Abstract](224)

*+*[PDF](295.8KB)

**Abstract:**

The paper deals with homogenization of an elliptic boundary value problem stated in a domain which consists of two connected components separated by a rapidly oscillating interface with a periodic microstructure, the interface being situated in a small neighbourhood of a hyperplane. At the interface we suppose the following transmission conditions: (i) the flux is continuous, (ii) the jump of a solution at the interface is proportional to the flux through the interface.

We derive the homogenized problem and effective transmission condition for different values of the ratio between the microstructure period and the amplitude of the interface oscillations, as well as for the different values of the mentioned proportionality coefficient.

*+*[Abstract](247)

*+*[PDF](168.6KB)

**Abstract:**

In this paper, we prove a higher integrability result for the gradient of a minimizer of a functional of the type

$I(\Omega , u)=\int_{\Omega}\sum_{i,j} a_{i,j} D_i u D_jv dx$

whose coefficient matrix $A(x)= ^tA(x)$ satisfies the anisotropic bounds

$\frac{|\xi |^2}{K(x)}\leq < A(x) \xi, \xi > \leq K(x) |\xi |^2\quad \forall \xi \in R^n,$ for a.e. $x\in \Omega,$

where $ K:\Omega \subset R^n \rightarrow [1,+\infty),$ a locally integrable function in $\Omega$, belongs to $A_2 \cap G_n$ and has a majorant $Q(x)\geq K(x)$ of finite mean,

limsup$_{R \rightarrow 0} \int_{B_R(x)} Q(y)dy < \infty $
*at every point * $x \in \Omega. $

*+*[Abstract](317)

*+*[PDF](235.7KB)

**Abstract:**

We study a $1$-capacitary type problem in $R^2$: given a set $E$, we minimize the perimeter (in the sense of De Giorgi) among all the sets containing $E$ (modulo $H^1$) and satisfying an indecomposability constraint (according to the definition by [1]. By suitably choosing the representant of the relevant set $E$, we show that a convexification process characterizes the minimizers.

As a consequence of our result we determine the $1$-capacity of (a suitable representant of) sets with finite perimeter in the plane.

*+*[Abstract](220)

*+*[PDF](124.1KB)

**Abstract:**

We consider a second order elliptic equation with measurable bounded coefficients

$ (a_{ij}(x)u_{x_i})_{x_j}+p(x)|x|^su^{-\sigma}=0, x\in\Omega \setminus \{ O\}, $

where $\sigma >0$, $s$ is any real number, and
$\Omega\subset R^n$, $n\ge 3$ is a bounded domain, which
contains the origin $O$.

The aim of this paper is to establish existence, nonexistence and
behavior of positive weak solutions near the isolated singularity
$O$.

*+*[Abstract](217)

*+*[PDF](258.2KB)

**Abstract:**

In this paper, a class of minimization problems, associated with the micromagnetics of thin films, is dealt with. Each minimization problem is distinguished by the thickness of the thin film, denoted by $ 0 < h < 1 $, and it is considered under spatial indefinite and degenerative setting of the material coefficients. On the basis of the fundamental studies of the governing energy functionals, the existence of minimizers, for every $ 0 < h < 1 $, and the 3D-2D asymptotic analysis for the observing minimization problems, as $ h \to 0 $, will be demonstrated in the main theorem of this paper.

*+*[Abstract](376)

*+*[PDF](221.0KB)

**Abstract:**

This paper is devoted to singular perturbation problems for first order equations. Under some coercivity and periodicity assumptions, we establish the uniform convergence and we provide an estimate of the rate of convergence, which we consider the main result of the paper.

We shall also show that our results apply to the homogenization problem for coercive and periodic equations. Finally, some examples arising in optimal control and differential games theory will be discussed.

*+*[Abstract](244)

*+*[PDF](157.1KB)

**Abstract:**

We develop a composite collocation approximation scheme for the numerical solution of nonlinear delay optimal control problems. For this purpose, we present an extension and also modification for the Gauss pseudospectral method using the hybrid of block-pulse functions and Lagrange polynomials based on the Legendre-Gauss points. In this respect, we derive the corresponding operational matrix of derivative according to the weak representation of derivative operator. In order to demonstrate the applicability, efficiency and accuracy of the proposed method, we examine two illustrative examples.

*+*[Abstract](210)

*+*[PDF](137.3KB)

**Abstract:**

We study planar homeomorphisms $f: \Omega\subset R^2 $ onto $\to \Omega' \subset R^2$, $f=(u,v)$, which are absolutely continuous on lines parallel to the axes (ACL) together with their inverse $f^{-1}$. The main result is that $u$ and $v$ have almost everywhere the same critical points. This generalizes a previous result ([6]) concerning bisobolev mappings. Moreover we construct an example of a planar ACL-homeomorphism not belonging to the Sobolev class $W_{l o c}^{1,1}$.

*+*[Abstract](193)

*+*[PDF](173.8KB)

**Abstract:**

We study the uniqueness of weak solutions for Dirichlet problems with variable exponent and non-standard growth conditions. First, we provide two uniqueness results under ellipticity type hypotheses. Next, we provide a uniqueness result when the operator driving the problem is in the form of the divergence of a monotone map. Finally, we derive a fourth uniqueness result under homogeneity type hypotheses, by means of a comparison result and approximation.

*+*[Abstract](287)

*+*[PDF](161.4KB)

**Abstract:**

We consider a Schrödinger-Poisson system with attractive selfinteractions and a multiple well external potential. We prove the existence of multiple breathing mode solutions bifurcating from the

*Hartree ground state*.

*+*[Abstract](257)

*+*[PDF](300.8KB)

**Abstract:**

Aim of this paper is to give some nonexistence results of nontrivial solutions for the following quasilinear elliptic equations with singular weights in $R^n\setminus \{0\}$

$ \Delta_p u+\mu|x|^{-\alpha}| u|^{a-2}u+\lambda | u|^{q-2}u+h(|x|)f(u) = 0 $ and

$ \Delta_p u+\mu|x|^{-\alpha}| u|^{p^*_\alpha-2}u+\lambda | u|^{q-2}u+h(|x|)f(u)= 0, $

where $1 < p < n$, $\alpha \in [0,p]$, $a \in [p,p^*_\alpha]$, $p_\alpha^*= p(n-\alpha)/(n-p)$, $\lambda, \mu \in R$ and $q \ge 1$, while $h: R^+ \to R^+_0$ and $f: R\to R$ are given continuous functions. The main tool for deriving nonexistence theorems for the equations is a Pohozaev--type identity. We first show that such identity holds true for weak solutions $u$ in $H^{1,p}(R^n)\cap C^1(R^n \setminus \{0\})$ of the first equation and for weak solutions $u$ in $D^{1,p}(R^n)\cap C^1(R^n \setminus \{0\})$ of the second equation. Then, under a suitable growth condition on $f$, we prove that every weak solution $u$ has the required regularity, so that the Pohozaev--type identity can be applied. From this identity we derive some nonexistence results, improving several theorems already appeared in the literature. In particular, we discuss the case when $h$ and $f$ are pure powers.

*+*[Abstract](241)

*+*[PDF](306.7KB)

**Abstract:**

We investigate Phragmèn-Lindelöf principles for viscosity solutions of fully nonlinear elliptic equations with possibly unbounded coefficients.

*+*[Abstract](344)

*+*[PDF](189.6KB)

**Abstract:**

The Möbius energy, defined 1991 by O'Hara, is the most prominent example of a knot energy. In this text we will focus on the regularity of local minimizers (within a prescribed knot class) whose arc-length parametrization was shown to be $C^{1,1}$ by Freedman, He, and Wang. Later on, He improved this result to $C^\infty$ regularity. In this text we will briefly outline the main ideas of these two steps which require completely different approaches involving techniques from geometry and analysis. Moreover we explain how to rigorously derive the first variation of the Möbius energy and fix a gap in He's treatise.

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