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European Mathematical Society Publishing House
2024-03-28 21:17:33
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Rendiconti Lincei - Matematica e Applicazioni
Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.
RLM
1120-6330
1720-0768
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10.4171/RLM
http://www.ems-ph.org/doi/10.4171/RLM
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2006)
26
2015
1
Lower semicontinuity for nonautonomous surface integrals
Virginia
De Cicco
Università di Roma La Sapienza, ROMA, ITALY
Semicontinuity, capacity, chain rule, $BV$ functions
Some lower semicontinuity results are established for nonautonomous surface integrals depending in a discontinuous way on the spatial variable. The proof of the semicontinuity results is based on some suitable approximations from below with appropriate functionals.
Calculus of variations and optimal control; optimization
1
21
10.4171/RLM/688
http://www.ems-ph.org/doi/10.4171/RLM/688
Transversally pseudoconvex semiholomorphic foliations
Samuele
Mongodi
Università di Roma 'Tor Vergata', ROMA, ITALY
Giuseppe
Tomassini
Scuola Normale Superiore, PISA, ITALY
Semiholomorphic foliations, CR geometry, complex spaces
A semiholomorphic foliations of type (n, d) is a differentiable real manifold $X$ of dimension $2n+d$, foliated by complex leaves of complex dimension $n$. The aim of the present note is to outline some results obtained in studying such spaces along the lines of the classical theory of complex spaces. Complete proofs will appear elsewhere.
Several complex variables and analytic spaces
Manifolds and cell complexes
23
36
10.4171/RLM/689
http://www.ems-ph.org/doi/10.4171/RLM/689
Some applications of numerosities in measure theory
Vieri
Benci
Università di Pisa, PISA, ITALY
Emanuele
Bottazzi
Università di Trento, TRENTO (TN), ITALY
Mauro
Di Nasso
Università di Pisa, PISA, ITALY
Non-Archimedean mathematics, measure theory, nonstandard analysis, numerosities
We present some applications of the notion of numerosity to measure theory, including the construction of a non-Archimedean model for the probability of infinite sequences of coin tosses.
Real functions
Measure and integration
Probability theory and stochastic processes
37
47
10.4171/RLM/690
http://www.ems-ph.org/doi/10.4171/RLM/690
On the critical polynomial of functionals related to $p$-area (for $1 < p < \infty$) and $p$-Laplace $(1 < p≤2)$ type operators
Silvia
Cingolani
Politecnico di Bari, BARI, ITALY
Marco
Degiovanni
Università Cattolica del Sacro Cuore, BRESCIA, ITALY
Giuseppina
Vannella
Politecnico di Bari, BARI, ITALY
$p$-area operator, $p$-Laplace operator, functionals with lack of smoothness, critical polynomial, Morse index
We consider a class of quasilinear elliptic equations whose principal part includes the $p$-area (for $1 < p < \infty$) and the $p$-Laplace (for $1 < p≤ 2)$ operator. For the critical points of the associated functional, we provide estimates of the corresponding critical polynomial.
Partial differential equations
Global analysis, analysis on manifolds
49
56
10.4171/RLM/691
http://www.ems-ph.org/doi/10.4171/RLM/691
On a Dirichlet problem with $p$-Laplacian and asymmetric nonlinearity
Salvatore
Marano
Università degli Studi di Catania, CATANIA, ITALY
Nikolaos
Papageorgiou
National Technical University of Athens, ATHENS, GREECE
$p$-Laplacian, asymmetric nonlinearity, critical groups, Morse identity
The existence of at least two nonnegative smooth solutions to a homogeneous Dirichlet problem with $p$-Laplacian and reaction ($p$–1)-linear, but asymmetric, at ±$\infty$ is investigated through variational and truncation techniques. The case $p = 2$ is separately examined, obtaining a third nontrivial smooth solution via Morse’s theory.
Partial differential equations
57
74
10.4171/RLM/692
http://www.ems-ph.org/doi/10.4171/RLM/692
Multiplicity of solutions of nonlinear scalar field equations
Riccardo
Molle
Università di Roma 'Tor Vergata', ROMA, ITALY
Donato
Passaseo
Università del Salento, LECCE, ITALY
Nonlinear scalar field equations, infinitely many solutions, variational methods
In this Note we present new multiplicity results for the solutions of nonlinear elliptic problems of the form $-\Delta u +a(x)u=|u|^{p-1}u$ in $\mathbb R^N$, $u\in H^1(\mathbb R^N)$, where $N\ge 2$, $p>1$, $p0$. In particular, we have infinitely many positive solutions when there exists $a_\infty>0$ such that $\lim_{|x|\to\infty}a(x)=a_\infty$ and $\lim_{|x|\to\infty}[a(x)-a_\infty]\,e^{\eta|x|}=+\infty$ $\forall\eta>0$.
Partial differential equations
75
82
10.4171/RLM/693
http://www.ems-ph.org/doi/10.4171/RLM/693
A coupling approach to Doob’s theorem
Alexei
Kulik
National Academy of Science of Ukraine, KYIV, UKRAINE
Michael
Scheutzow
Technische Universität Berlin, BERLIN, GERMANY
Markov process, invariant measure, coupling, convergence of transition probabilities, total variation distance
We provide a coupling proof of Doob’s theorem which says that the transition probabilities of a regular Markov process which has an invariant probability measure $\mu$ converge to $\mu$ in the total variation distance. In addition we show that non-singularity (rather than equivalence) of the transition probabilities suffices to ensure convergence of the transition probabilities for $\mu$-almost all initial conditions.
Probability theory and stochastic processes
Dynamical systems and ergodic theory
83
92
10.4171/RLM/694
http://www.ems-ph.org/doi/10.4171/RLM/694
Sets of admissible shifts of convex measures
Lavrentin
Arutyunyan
Moscow State University, MOSCOW, RUSSIAN FEDERATION
Egor
Kosov
Moscow State University, MOSCOW, RUSSIAN FEDERATION
Convex measure, logarithmically concave measure, admissible shift
We prove that for any convex probability measure on a linear space the set of its non-singular shifts is convex and the set of its equivalent shifts is a linear subspace.
Measure and integration
Functional analysis
Probability theory and stochastic processes
93
98
10.4171/RLM/695
http://www.ems-ph.org/doi/10.4171/RLM/695
A survey on the existence of isoperimetric sets in the space $\mathbb R^N$ with density
Aldo
Pratelli
Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
Isoperimetric problem, existence of optimal sets, perimeter with density
The aim of this survey is to give a precise idea of the recent results on existence of isoperimetric sets in $\mathbb R^N$ with density. We will mainly focus on the overall ideas, leaving away some technical details of the proofs, which can be found in the cited papers. No previous knowledge on the subject is assumed from the reader. This survey originates from a talk of the author at the conference ‘‘New Trends in Nonlinear PDE’s’’ held at the Accademia dei Lincei on November 26th, 2013. I wish to dedicate this paper to Carlo Sbordone, because of his recent 65th birthday, and to Ula, because she will become my wife in a few days.
Calculus of variations and optimal control; optimization
99
118
10.4171/RLM/696
http://www.ems-ph.org/doi/10.4171/RLM/696
2
The Neumann eigenvalue problem for the $\infty$-Laplacian
L.
Esposito
Università di Salerno, FISCIANO (SA), ITALY
Bernd
Kawohl
Universität Köln, KÖLN, GERMANY
Carlo
Nitsch
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Cristina
Trombetti
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Neumann eigenvalues, viscosity solutions, infinity Laplacian
The first nontrivial eigenfunction of the Neumann eigenvalue problem for the $p$-Laplacian, suitably normalized, converges to a viscosity solution of an eigenvalue problem for the $\infty$-Laplacian as $p \to \infty$. We show among other things that the limiting eigenvalue, at least for convex sets, is in fact the first nonzero eigenvalue of the limiting problem. We then derive a number of consequences, which are nonlinear analogues of well-known inequalities for the linear (2-)Laplacian.
Partial differential equations
119
134
10.4171/RLM/697
http://www.ems-ph.org/doi/10.4171/RLM/697
Maximal regularity for gradient systems with boundary degeneracy
Piermarco
Cannarsa
Università di Roma, ROMA, ITALY
Giuseppe
Da Prato
Scuola Normale Superiore, PISA, ITALY
Giorgio
Metafune
Università del Salento, LECCE, ITALY
Diego
Pallara
Università degli Studi di Lecce, LECCE, ITALY
Degenerate elliptic operators, di¤usion processes, semigroups of operators, gradient systems, invariant measure
We study a class of elliptic operators $L$ that degenerate at the boundary of a bounded open set ${\mathcal O}\subset \mathbb R^d$ and possess a symmetrizing invariant measure $\mu$. Such operators are associated with diffusion processes in $\mathcal O$ which are invariant for time reversal. After showing that the corresponding elliptic equation $\lambda\varphi -L\varphi=f$ has a unique weak solution for any $\lambda>0$ and $f\in L^2(\mathcal O,\mu)$, we obtain new results for the characterization of the domain of $L$.
Partial differential equations
Operator theory
Probability theory and stochastic processes
135
149
10.4171/RLM/698
http://www.ems-ph.org/doi/10.4171/RLM/698
Functions determining locally solid topological Riesz spaces continuously embedded in $L^0$
Paola
Cavaliere
Università di Salerno, FISCIANO (SA), ITALY
Paolo
de Lucia
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Anna
De Simone
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Measurable function, convergence in measure, topological Riesz space, quasitriangular function, quasi-norm, embedding
We present a class of non-negative functions, acting on a solid vector subspace $X$ of $L^0$, enjoying the following property: each member of the class determines on $X$ a locally solid topological Riesz space structure which is continuously embedded into $L^0$. These functions are neither necessarily monotone, nor subadditive. Special instances are provided by function norms and quasi-norms on $X$.
Functional analysis
Measure and integration
151
160
10.4171/RLM/699
http://www.ems-ph.org/doi/10.4171/RLM/699
On doubly nonlocal fractional elliptic equations
Giovanni
Molica Bisci
Università degli Studi Mediterranea de Reggio Calabria, REGGIO CALABRIA, ITALY
Dušan
Repovš
University of Ljubljana, LJUBLJANA, SLOVENIA
Nonlocal problems, fractional equations, Mountain Pass Theorem
This work is devoted to the study of the existence of solutions to nonlocal equations involving the fractional Laplacian. These equations have a variational structure and we find a nontrivial solution for them using the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. In addition, we require rather general assumptions on the local operator. As far as we know, this result is new and represent a fractional version of a classical theorem obtained working with Laplacian equations.
Operator theory
Integral equations
161
176
10.4171/RLM/700
http://www.ems-ph.org/doi/10.4171/RLM/700
Moser-Trudinger inequality in grand Lebesgue space
Robert
Černý
Charles University, PRAGUE 8, CZECH REPUBLIC
Grand Lebesgue space, Sobolev spaces, embedding theorems, sharp constants, Moser-Trudinger inequality
Let $n \in \mathbb N, n ≥ 2$ and let $\Omega \subset \mathbb R^n$ be a bounded domain. We study sharp constants for the Moser-Trudinger inequality in the Sobolev-type space $W_0L^{n)}(\Omega)$, where $L^{n)} \Omega$ is so called grand $L^n$ space introduced in [9]. In particular, we obtain our results with respect to two quantities introduced in [8].
Functional analysis
Real functions
177
188
10.4171/RLM/701
http://www.ems-ph.org/doi/10.4171/RLM/701
Some remarks on a linearized Schrödinger equation
Giorgio
Busoni
Universita di Firenze, FIRENZE, ITALY
Laura
Prati
Universita degli Studi di Firenze, FIRENZE, ITALY
Grand Lebesgue space, Sobolev spaces, embedding theorems, sharp constants, Moser-Trudinger inequality
In the present paper we propose four systems of linear Partial Differential Equations that can be deduced from the nonlinear Schrödinger equation for the propagation of light in optical fibers in the frame of the recently-proposed Combined Regular-Logarithmic Perturbation method. The unknown function in the Schrödinger equation is the optical field envelope; it is a complex-valued function. Following the Combined Regular-Logarithmic Perturbation method, proposed by Secondini, Forestieri and Menyuk, we look for complex solutions of the Schrödinger equation in the form of a perturbed continuous wave that relates three unknown real-valued functions. Since the Schrödinger equation is complex, we split it into two real equations, both in the three real unknowns. We linearize these two equations and add a third linear equation that relates the same three unknown quantities. We propose four different choices for the third equation, therefore we obtain four di¤erent real systems of linear Partial Di¤erential Equations and we analyze the corresponding systems of Ordinary Differential Equations for the Fourier transforms of the unknowns. One of the four systems we obtain is equivalent to that studied by the quoted authors. We add to it other three choices that could be useful to model di¤erent situations. Again, we consider the real part of the Ordinary Differential Equations and we present solutions in recursive form. We also suggest solutions for the complex-valued Fourier transforms by using Bessel functions.
Partial differential equations
Ordinary differential equations
189
213
10.4171/RLM/702
http://www.ems-ph.org/doi/10.4171/RLM/702
A failing in the Calderon-Zygmund theory of Dirichlet problems for linear equations with discontinuous coefficients
Lucio
Boccardo
Università di Roma La Sapienza, ROMA, ITALY
Failing in the Calderon-Zygmund theory, Dirichlet problems, linear equations with discontinuous coefficients
This note is devoted to the Calderon-Zygmund theory for linear differential operators with discontinuous coefficients. It is known that the theory holds if the datum $f(x)$, in (1.1) belongs to the Lebesgue space $L^m \Omega$, with $1 < m < \frac {2N}{N+2}$ (see [6]). In this paper we prove that the theory fails if $m >\frac {N}{2}$.
Partial differential equations
215
221
10.4171/RLM/703
http://www.ems-ph.org/doi/10.4171/RLM/703
New approximations of the total variation and filters in Imaging
Haim
Brezis
Rutgers University, Hill Center, Busch Campus, PISCATAWAY, UNITED STATES
Total variation, bounded variation, non-local functional, non-convex functional, $\Gamma$-convergence
In this paper I present several results concerning the approximation of the BV-norm by non-local functionals. Some of these functionals are convex, others are non-convex. The mode of convergence introduces mysterious novelties and numerous problems remain open. The original motivation comes from Image Processing.
Partial differential equations
Real functions
Functional analysis
223
240
10.4171/RLM/704
http://www.ems-ph.org/doi/10.4171/RLM/704
3
Linear equation with data in non standard spaces
Jean-Michel
Rakotoson
Université de Poitiers, FUTUROSCOPE CHASSENEUIL CEDEX, FRANCE
Mathematical analysis, functional analysis, partial di¤erential equation
Given a finite family of Banach function spaces $V_\alpha$ over a bounded set $\Omega$, $V=\prod_\alpha V_\alpha$, and let $T$ an element of the dual of the Sobolev space $W^2V$. We discuss the existence, uniqueness and regularity of the solution of the linear equation $Lu=T$ under the Dirichlet or Neumann condition on the boundary of $\Omega$. Our results extend recent works on very weak solution with data in weighted distance space or Lorentz space.
Functional analysis
Partial differential equations
241
262
10.4171/RLM/705
http://www.ems-ph.org/doi/10.4171/RLM/705
The Skolem-Abouzaïd theorem in the singular case
Boris
Bartolome
Enteleia Tech, AUREVILLE, FRANCE
Skolem-Abouzaïd, Puiseaux series, lgcd, heights
Let ${F(X,Y)\in\mathbb Q[X,Y]}$ be a $\mathbb Q$-irreducible polynomial. In 1929, Skolem [13] proved a result allowing explicit bounding of the solutions of $F(X,Y)=0$ such that $\mathrm {gcd} (X,Y)=d$ in terms of the coefficients of $F$ and $d$. In 2008, Abouzaïd [1] generalized this result by working with arbitrary algebraic numbers and by obtaining an asymptotic relation between the heights of the coordinates and their logarithmic gcd. However, he imposed the condition that $(0,0)$ be a non-singular point of the plane curve $F(X,Y)=0$. In this paper, we remove this constraint.
Number theory
263
289
10.4171/RLM/706
http://www.ems-ph.org/doi/10.4171/RLM/706
A Sobolev non embedding
Petru
Mironescu
Université Lyon 1, VILLEURBANNE CEDEX, FRANCE
Björn
Schmalfuss
Friedrich-Schiller-Universität Jena, JENA, GERMANY
Sobolev and Slobodeskii spaces, embeddings, lacunary series, wavelets
If $\Omega$ is a bounded domain in $\mathbb R^n$, $1 ≤ q < p ≤ \infty$ and $s=0, 1, 2,\ldots$, then we clearly have $W^{s,p}(\Omega)\subset W^{s,q}(\Omega)$. We prove that this property does not hold when $s$ in not an integer.
Functional analysis
291
298
10.4171/RLM/707
http://www.ems-ph.org/doi/10.4171/RLM/707
A note on compactness properties of the singular Toda system
Luca
Battaglia
SISSA, TRIESTE, ITALY
Gabriele
Mancini
SISSA, TRIESTE, ITALY
Toda system, compactness of solutions, blow-up analysis, mass quantization
In this note, we consider blow-up for solutions of the $SU(3)$ Toda system on a compact surface $\Sigma$. In particular, we give a complete proof of the compactness result stated by Jost, Lin and Wang in [11] and we extend it to the case of singularities. This is a necessary tool to find solutions through variational methods.
Partial differential equations
299
307
10.4171/RLM/708
http://www.ems-ph.org/doi/10.4171/RLM/708
A sharp quantitative isoperimetric inequality in higher codimension
Verena
Bögelein
Universität Salzburg, SALZBURG, AUSTRIA
Frank
Duzaar
Universität Erlangen-Nünberg, ERLANGEN, GERMANY
Nicola
Fusco
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Isoperimetric inequality, area minimizing currents, stability
We establish a quantitative isoperimetric inequality in higher codimension. In particular, we prove that for any closed $(n-1)$-dimensional manifold $\Gamma$ in $\mathbb R^{n+k}$ the following inequality $$\mathbf D(\Gamma)\ge C \mathbf d^2(\Gamma)$$ holds true. Here, $\mathbf D(\Gamma)$ stands for the isoperimetric gap of $\Gamma$, i.e. the deviation in measure of $\Gamma$ from being a round sphere and $\mathbf d(\Gamma )$ denotes a natural generalization of the Fraenkel asymmetry index of $\Gamma$ to higher codimensions.
Calculus of variations and optimal control; optimization
Convex and discrete geometry
309
362
10.4171/RLM/709
http://www.ems-ph.org/doi/10.4171/RLM/709
4
Henstock multivalued integrability in Banach lattices with respect to pointwise non atomic measures
Antonio
Boccuto
Università degli Studi di Perugia, PERUGIA, ITALY
Domenico
Candeloro
Università degli Studi di Perugia, PERUGIA, ITALY
Anna Rita
Sambucini
Università degli Studi di Perugia, PERUGIA, ITALY
Banach lattices, Henstock integral, McShane integral, multivalued integral, pointwise non atomic measures
Henstock-type integrals are considered, for multifunctions taking values in the family of weakly compact and convex subsets of a Banach lattice $X$. The main tool to handle the multivalued case is a Rådström-type embedding theorem established by C. C. A. Labuschagne, A. L. Pinchuck, C. J. van Alten in 2007. In this way the norm and order integrals reduce to that of a single-valued function taking values in an $M$-space, and new proofs are deduced for some decomposition results recently stated in two recent papers by Di Piazza and Musiał based on the existence of integrable selections.
Measure and integration
Category theory; homological algebra
Functional analysis
363
383
10.4171/RLM/710
http://www.ems-ph.org/doi/10.4171/RLM/710
A Harnack’s inequality and Hölder continuity for solutions of mixed type evolution equations
Fabio
Paronetto
Università degli Studi di Padova, PADOVA, ITALY
Mixed type equations, Harnack’s inequality, Hölder-continuity
We define a homogeneous parabolic De Giorgi classes of order 2 which suits a mixed type class of evolution equations whose simplest example is $\mu (x) \frac{\partial u}{\partial t} - \Delta u = 0$ where $\mu$ can be positive, null and negative, so that elliptic-parabolic and forward-backward parabolic equations are included. For functions belonging to this class we prove local boundedness and show a Harnack inequality which, as by-products, gives Hölder-continuity, in particular in the interface $I$ where $\mu$ change sign, and a maximum principle.
Partial differential equations
385
395
10.4171/RLM/711
http://www.ems-ph.org/doi/10.4171/RLM/711
A density property for fractional weighted Sobolev spaces
Serena
Dipierro
University of Edinburgh, EDINBURGH, UNITED KINGDOM
Enrico
Valdinoci
Università degli Studi di Milano, MILANO, ITALY
Weighted fractional Sobolev spaces, density properties
In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.
Functional analysis
Partial differential equations
397
422
10.4171/RLM/712
http://www.ems-ph.org/doi/10.4171/RLM/712
On the measure of Lagrangian invariant tori in nearly-integrable mechanical systems
Luca
Biasco
Università degli studi Roma Tre, ROMA, ITALY
Luigi
Chierchia
Università degli Studi Roma Tre, ROMA, ITALY
Nearly-integrable Hamiltonian systems, normal forms, KAM theory
Consider an $n$-degrees-of-freedom real-analytic mechanical system with potential $\epsilon f = \epsilon f (x)$, $x$ being a $n$-dimensional angle variable. Then, for ‘‘general’’ potentials $f$’s and $\epsilon$ small enough, the Liouville measure of the complementary of invariant tori is smaller than $\epsilon |\mathrm {ln} \epsilon |^a$ (for a suitable $a > 0$).
Dynamical systems and ergodic theory
423
432
10.4171/RLM/713
http://www.ems-ph.org/doi/10.4171/RLM/713
$L^p$ theory for fractional gradient PDE with $VMO$ coefficients
Armin
Schikorra
Universität Basel, BASEL, SWITZERLAND
Tien-Tsan
Shieh
National Central University, CHUNG LI, TAIWAN
Daniel
Spector
National Chiao Tung University, HSINCHU, TAIWAN
Fractional elliptic PDE, VMO coefficients, commutators
In this paper, we prove $L^p$ estimates for the fractional derivatives of solutions to elliptic fractional partial differential equations whose coefficients are $VMO$. In particular, our work extends the optimal regularity known in the second order elliptic setting to a spectrum of fractional order elliptic equations.
Partial differential equations
433
443
10.4171/RLM/714
http://www.ems-ph.org/doi/10.4171/RLM/714
Existence, uniqueness and behaviour of solutions for a nonlinear diffusion equation with third type boundary value condition
Fatma Gamze
Duzgun
Hacettepe University, BEYTEPE, ANKARA, TURKEY
Kamal
Soltanov
Hacettepe University, BEYTEPE, ANKARA, TURKEY
Nonlinear diffusion equations, Robin boundary condition, non-local effect, existence and uniqueness, behavior of solution, absorbing set
In this work, we investigate a mixed problem with boundary condition of third type for a nonlinear diffusion equation having nonlocal term. Existence and uniqueness of a solution of the posed problem are proved under fairly general conditions. Moreover, we obtain some results on the behaviour of the solution and the existence of an absorbing set for the problem under consideration.
Partial differential equations
445
463
10.4171/RLM/715
http://www.ems-ph.org/doi/10.4171/RLM/715
Quasi-filling fractal layers
Raffaela
Capitanelli
Università di Roma La Sapienza, ROMA, ITALY
Maria Agostina
Vivaldi
Università di Roma La Sapienza, ROMA, ITALY
Fractal fibers, singular elliptic operators, variational convergence
We consider second order transmission problems across Koch-type curves formulated as boundary value problems for elliptic operators in a quasi-filling geometry for the fibers. We use a variational approach and the M-convergence methods. We prove that the solution of the transmission problem across a Koch-type curve is the limit of the solutions of suitable second order transmission problems across polygonal curves.
Partial differential equations
Measure and integration
465
473
10.4171/RLM/716
http://www.ems-ph.org/doi/10.4171/RLM/716
Regularity results for non-autonomous variational integrals with discontinuous coefficients
Antonia
Passarelli di Napoli
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Elliptic systems, discontinuous coeffcients, higher differentiability
We investigate the regularity propertiesof local minimizers of non autonomous convex integral functionals of the type $$\mathcal{F}(u; \Omega):= \int_{\Omega} f (x, Du) \ dx ,$$ with $p$-growth into the gradient variable and discontinuous dependence on the $x$ variable. We prove a higher differentiability result for local minimizers of the functional $\mathcal{F}(u; \Omega)$ assuming that the function that measures the oscillation of the integrand with respect to the $x$ variable belongs to a suitable Sobolev space.
Calculus of variations and optimal control; optimization
475
496
10.4171/RLM/717
http://www.ems-ph.org/doi/10.4171/RLM/717