Halidias Nikolaos

Elliptic problems with nonmonotone discontinuities at resonance (Erratum)

We study the existence and multiplicity of solutions for a class
of quasilinear elliptic problem in exterior domain with Neumann
boundary conditions.

Claudianor O. Alves

Paulo C. Carri&#xE3;o

Everaldo S. Medeiros

Multiplicity of solutions for a class of quasilinear problem in
exterior domains with Neumann conditions

We first introduce a modified proximal point algorithm for
maximal monotone operators in a Banach space. Next, we obtain a
strong convergence theorem for resolvents of maximal monotone
operators in a Banach space which generalizes the previous result
by Kamimura and Takahashi in a Hilbert space. Using this result,
we deal with the convex minimization problem and the variational
inequality problem in a Banach space.

Fumiaki  Kohsaka

Wataru  Takahashi

Strong convergence of an iterative sequence for maximal monotone operators in a Banach space

We consider a two-phase system mainly in three dimensions and we
examine the coarsening of the spatial distribution, driven by the
reduction of interface energy and limited by diffusion as
described by the quasistatic Stefan free boundary problem. Under
the appropriate scaling we pass rigorously to the limit by taking
into account the motion of the centers and the deformation of the
spherical shape. We distinguish between two different cases and we
derive the classical mean-field model and another continuum limit
corresponding to critical density which can be related to a
continuity equation obtained recently by Niethammer andOtto. 
So, the theory of Lifshitz, Slyozov, and Wagner is improved by taking
into account the geometry of the spatial distribution.

Nicholas D. Alikakos

Giorgio  Fusco

Georgia  Karali

Continuum limits of particles interacting via diffusion

We prove an existence result for solution to a class of nonlinear degenerate elliptic equation associated with a class of partial
differential operators of the form <mml:math alttext="$Lu(x) = \sum_{i,j=1}^nD_j (a_{ij}(x)D_iu(x))$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mo>&#x2211;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, with <mml:math alttext="$D_j =\partial/\partial{x_j}$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>D</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>&#x2202;</mml:mo><mml:mo>/</mml:mo><mml:mrow><mml:mo>&#x2202;</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:math>, where <mml:math alttext="$a_{ij}: \Omega \rightarrow \mathbb{R}$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mi>&#x3A9;</mml:mi><mml:mo>&#x2192;</mml:mo><mml:mi>&#x211D;</mml:mi></mml:math> are functionssatisfying suitable hypotheses.

Albo Carlos Cavalheiro

Solvability of nonlinear Dirichlet problem for a class of degenerate elliptic equations

In the case of <mml:math alttext="$K \neq \overline{D(A)}$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>K</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="true">&#xAF;</mml:mo></mml:mover></mml:mrow></mml:math>, we study Cauchy problems
and periodic problems for nonlinear evolution equation <mml:math alttext="$ u(t) \in K$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2208;</mml:mo><mml:mi>K</mml:mi></mml:math>, <mml:math alttext="$u'(t)+Au(t)\ni f(t,u(t))$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>u</mml:mi><mml:mo>&#x2032;</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>&#x220B;</mml:mi><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>, <mml:math alttext="$0 \leq t \leq T$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:mi>t</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mi>T</mml:mi></mml:math>, where <mml:math alttext="$A$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> isa
maximal monotone operator on a Hilbert space <mml:math alttext="$H$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math>, <mml:math alttext="$K$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> is a
closed, convex subset of <mml:math alttext="$H$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math>, <mml:math alttext="$V$" id="E9" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>V</mml:mi></mml:math> is a subspace of <mml:math alttext="$H$" id="E10" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math>, and <mml:math alttext="$f: [0,T] \times (K \cap V) \rightarrow H$" id="E11" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>&#xD7;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>K</mml:mi><mml:mo>&#x2229;</mml:mo><mml:mi>V</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2192;</mml:mo><mml:mi>H</mml:mi></mml:math> is of Carath&#xE9;odory
type.

Norimichi Hirano

Naoki Shioji

Invariant sets for nonlinear evolution equations, Cauchy problems
and periodic problems

Nonselfadjoint boundary value problems for second-order differential equations on a finite interval with nonintegrable singularities inside the interval are considered under additional sewing conditions for solutions at the singular point. We study properties of the spectrum, prove the completeness of eigen- and associated functions, and investigate the inverse problem of recovering the boundary value problem from its spectral characteristics.

Vjacheslav Anatoljevich Yurko

Spectral analysis for differential operators with singularities

We establish nonexistence results to systems of differential inequalities on the <mml:math alttext="$(2N+1)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>N</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>-Heisenberg group. The systems considered here are of the type <mml:math alttext="(ES$_m$)" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mtext>ES</mml:mtext></mml:mrow><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>. These nonexistence results hold for <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> less than critical exponents which depend on <mml:math alttext="$p_i$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math> and <mml:math alttext="$\gamma_i$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>&#947;</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>, <mml:math alttext="$1\leq i \leq m$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#8804;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#8804;</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:math>. Our results improve the known estimates of the critical exponent.

Abdallah  El Hamidi

Mokhtar  Kirane

Nonexistence results of solutions to systems of semilinear differential inequalities on the Heisenberg group

We propose another extension of Orlicz-Sobolev spaces to metric spaces based on the concepts of the <mml:math alttext="$\Phi$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#934;</mml:mi></mml:math>-modulus and <mml:math alttext="$\Phi$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#934;</mml:mi></mml:math>-capacity. The resulting space <mml:math alttext="$N_{\Phi}^{1}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi><mml:mi>&#934;</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> is a Banach space. The relationship between <mml:math alttext="$N_{\Phi}^{1}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi><mml:mi>&#934;</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> and <mml:math alttext="$M_{\Phi}^{1}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>M</mml:mi><mml:mi>&#934;</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> (the first extension defined in A&#239;ssaoui (2002)) is studied. We also explore and compare different definitions of capacities and give a criterion under which <mml:math alttext="$N_{\Phi}^{1}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi><mml:mi>&#934;</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> is strictly smaller than the Orlicz space <mml:math alttext="$\mathbf{L}_{\Phi}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="bold">L</mml:mi><mml:mi>&#934;</mml:mi></mml:msub></mml:math>.

Noureddine  A&#239;ssaoui

Another extension of Orlicz-Sobolev spaces to metric spaces

About this journal

Abstract and Applied Analysis publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.

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Chief Editor, Dr Wong, is an associate professor at Nanyang Technological University, Singapore. Her research interests include differential equations, difference equations, integral equations, and numerical mathematics.

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Abstract and Applied Analysis - Table of contents 2004

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Abstract and Applied Analysis publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.

Abstract and Applied Analysis