# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

June 2018 , Volume 11 , Issue 3

Issue on recent progresses in the theory of nonlinear nonlocal problems

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2018, 11(3): i-i doi: 10.3934/dcdss.201803i +[Abstract](3313) +[HTML](210) +[PDF](82.2KB)
Abstract:
2018, 11(3): 357-377 doi: 10.3934/dcdss.2018020 +[Abstract](4184) +[HTML](348) +[PDF](472.2KB)
Abstract:

Many structural materials, which are preferred for the developing of advanced constructions, are inhomogeneous ones. Composite materials have complex internal structure and properties, which make them to be more effectual in the solution of special problems required for civil and environmental engineering. As a consequence of this internal heterogeneity, they exhibit complex mechanical properties. In this work, the analysis of some features of the behavior of composite materials under different loading conditions is carried out. The dependence of nonlinear elastic response of composite materials on loading conditions is studied. Several approaches to model elastic nonlinearity such as different stiffness for particular type of loadings and nonlinear shear stress–strain relations are considered. Instead of a set of constant anisotropy coefficients, the anisotropy functions are introduced. Eventually, the combined constitutive relations are proposed to describe simultaneously two types of physical nonlinearities. The first characterizes the nonlinearity of shear stress–strain dependency and the latter determines the stress state susceptibility of material properties. Quite satisfactory correlation between the theoretical dependencies and the results of experimental studies is demonstrated, as described in [2,3] as well as in the references therein.

2018, 11(3): 379-389 doi: 10.3934/dcdss.2018021 +[Abstract](6670) +[HTML](445) +[PDF](418.1KB)
Abstract:

The content of this paper is at the interplay between function spaces $L^{p(x)}$ and $W^{k, p(x)}$ with variable exponents and fractional Sobolev spaces $W^{s, p}$. We are concerned with some qualitative properties of the fractional Sobolev space $W^{s, q(x), p(x, y)}$, where $q$ and $p$ are variable exponents and $s∈ (0, 1)$. We also study a related nonlocal operator, which is a fractional version of the nonhomogeneous $p(x)$-Laplace operator. The abstract results established in this paper are applied in the variational analysis of a class of nonlocal fractional problems with several variable exponents.

2018, 11(3): 391-424 doi: 10.3934/dcdss.2018022 +[Abstract](4062) +[HTML](206) +[PDF](572.5KB)
Abstract:

We obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional $p-$Laplacian operator and nonlinearities at critical growth.

2018, 11(3): 425-440 doi: 10.3934/dcdss.2018023 +[Abstract](3240) +[HTML](189) +[PDF](396.0KB)
Abstract:

We consider a class of nonlocal generalized perimeters which includes fractional perimeters and Riesz type potentials. We prove a general isoperimetric inequality for such functionals, and we discuss some applications. In particular we prove existence of an isoperimetric profile, under suitable assumptions on the interaction kernel.

2018, 11(3): 441-463 doi: 10.3934/dcdss.2018024 +[Abstract](3701) +[HTML](204) +[PDF](488.2KB)
Abstract:

We establish existence and qualitative properties of solutions to the fractional Allen-Cahn equation, which vanish on the Simons cone and are even with respect to the coordinate axes. These solutions are called saddle-shaped solutions.

More precisely, we prove monotonicity properties, asymptotic behaviour, and instability in dimensions $2m=4, 6$. We extend to any fractional power $s$ of the Laplacian, some results obtained for the case $s=1/2$ in [19].

The interest in the study of saddle-shaped solutions comes in connection with a celebrated De Giorgi conjecture on the one-dimensional symmetry of monotone solutions and of minimizers for the Allen-Cahn equation. Saddle-shaped solutions are candidates to be (not one-dimensional) minimizers in high dimension, a property which is not known to hold yet.

2018, 11(3): 465-476 doi: 10.3934/dcdss.2018025 +[Abstract](4051) +[HTML](186) +[PDF](397.5KB)
Abstract:

We discuss some recent developments in the theory of free boundary problems, as obtained in a series of papers in collaboration with L. Caffarelli, A. Karakhanyan and O. Savin.

The main feature of these new free boundary problems is that they deeply take into account nonlinear energy superpositions and possibly nonlocal functionals.

The nonlocal parameter interpolates between volume and perimeter functionals, and so it can be seen as a fractional counterpart of classical free boundary problems, in which the bulk energy presents nonlocal aspects.

The nonlinear term in the energy superposition takes into account the possibility of modeling different regimes in terms of different energy levels and provides a lack of scale invariance, which in turn may cause a structural instability of minimizers that may vary from one scale to another.

2018, 11(3): 477-491 doi: 10.3934/dcdss.2018026 +[Abstract](4322) +[HTML](181) +[PDF](419.4KB)
Abstract:

We define and study the fractional Laplacian and the fractional perimeter of a set in Carnot groups and we compare the perimeter with the asymptotic behaviour of the fractional heat semigroup.

2018, 11(3): 493-509 doi: 10.3934/dcdss.2018027 +[Abstract](3365) +[HTML](161) +[PDF](416.6KB)
Abstract:

In this paper we study some boundary operators of a class of Bessel-type Littlewood-Paley extensions whose prototype is

In particular, we show that with a logarithmic scaling one can capture the failure of analyticity of these extensions in the limiting cases \begin{document} $s=k ∈ \mathbb{N}$ \end{document}.

2018, 11(3): 511-532 doi: 10.3934/dcdss.2018028 +[Abstract](3787) +[HTML](199) +[PDF](478.3KB)
Abstract:

We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory.

2018, 11(3): 533-545 doi: 10.3934/dcdss.2018029 +[Abstract](4352) +[HTML](267) +[PDF](468.4KB)
Abstract:

In this paper, we are concerned with the following fractional Kirchhoff equation

where \begin{document} $N>2s$ \end{document}, \begin{document} $a, b, \lambda, μ>0$ \end{document}, \begin{document} $s∈(0, 1)$ \end{document} and \begin{document} $Ω$ \end{document} is a bounded open domain with continuous boundary. Here \begin{document} $(-Δ)^s$ \end{document} is the fractional Laplacian operator. For \begin{document} $2<q≤q\min\{4, 2_s^*\}$ \end{document}, we prove that if \begin{document} $b$ \end{document} is small or \begin{document} $μ$ \end{document} is large, the problem above admits multiple solutions by virtue of a linking theorem due to G. Cerami, D. Fortunato and M. Struwe [7, Theorem 2.5].

2018, 11(3): 547-559 doi: 10.3934/dcdss.2018030 +[Abstract](3629) +[HTML](182) +[PDF](400.5KB)
Abstract:

Given \begin{document} $L≥1$ \end{document}, we discuss the problem of determining the highest \begin{document} $α=α(L)$ \end{document} such that any solution to a homogeneous elliptic equation in divergence form with ellipticity ratio bounded by \begin{document} $L$ \end{document} is in \begin{document} $C^α_{\rm loc}$ \end{document}. This problem can be formulated both in the classical and non-local framework. In the classical case it is known that \begin{document} $α(L)≳ {\rm exp}(-CL^β)$ \end{document}, for some \begin{document} $C, β≥q$ \end{document} depending on the dimension \begin{document} $N≥q$ \end{document}. We show that in the non-local case, \begin{document} $α(L)≳ L^{-1-δ}$ \end{document} for all \begin{document} $δ>0$ \end{document}.

2018, 11(3): 561-576 doi: 10.3934/dcdss.2018031 +[Abstract](4095) +[HTML](219) +[PDF](466.1KB)
Abstract:

By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends the literature for the \begin{document} $N$ \end{document}-Laplacian operator.

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