Mathematics

We establish the existence and find some qualitative properties of open sets that minimize functionals of the form F( λ 1 (Ω;β),?? λ k (Ω;β)) under measure constraint on Ω , where λ i (Ω;β) designates the i -th eigenvalue of the Laplace operator on Ω with Robin boundary conditions of parameter β>0 . Moreover, we show that minimizers of λ k (Ω;β) for k?? verify the conjecture λ k (Ω;β)= λ k?? (Ω;β) in dimension three and more.

In this note we study analytically and numerically the existence and stability of standing waves for one dimensional nonlinear Schrödinger equations whose nonlinearities are the sum of three powers. Special attention is paid to the curves of non-existence and curves of stability change on the parameter planes.

The present work constitutes a first step towards establishing a systematic framework for treating variational problems that depend on a given input function through a mixture of its derivatives of different orders in different directions. For a fixed vector a:=( a 1 ,?? a N )??N N and u: R N ?�Ω�? R n we denote by ??a u:=( ??α u ) ?��? a ?? ??1 the matrix whose i -th row is composed of derivatives ??α u i of the i -th component of the map u , and where the multi-indices α satisfy ?��? a ?? ?? ??N j=1 α j a j =1 . We study functionals of the form W a,p (Ω; R n )?�u????Ω F( ??a u(x))dx, where W a,p (Ω; R n ) is an appropriate Sobolev space of mixed smoothness and F is the integrand. We study existence of minimisers of such functionals under prescribed Dirichlet boundary conditions. We characterise coercivity, lower semicontiuity, and envelopes of relaxation of such functionals, in terms of an appropriate generalisation of Morrey's quasiconvexity.

In this paper we study quasilinear elliptic equations driven by the double phase operator and a right-hand side which has the combined effect of a singular and of a parametric term. Based on the Nehari manifold method we are going to prove the existence of at least two weak solutions for such problem when the parameter is sufficiently small.

We establish the existence of solutions to the Cauchy problem for a large class of nonlinear parabolic equations including fractional semilinear parabolic equations, higher-order semilinear parabolic equations, and viscous Hamilton-Jacobi equations by using the majorant kernel introduced in [K. Ishige, T. Kawakami, and S. Okabe, Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), 1185--1209].

In this work we will focus on the existence of weak solutions for a system describing a general compressible viscous fluid in the case of the pressure being a linear function of the density and the viscous stress tensor being a non-linear function of the symmetric velocity gradient. More precisely, we will first prove the existence of dissipative solutions and study under which conditions it is possible to guarantee the existence of weak solutions.

In this paper we analyze the three-dimensional Peterlin viscoelastic model. By means of a mixed Galerkin and semigroup approach we prove the existence of a weak solutions. Further combining parabolic regularity with the relative energy method we derive a conditional weak-strong uniqueness result.

In this paper we extend a recent work \cite{HuyXuan2020} to study the forward asymptotically almost periodic (AAP-) mild solution of Navier-Stokes equation on the real hyperbolic manifold H d (R) with dimension d?? . Using the dispertive and smoothing estimates for Stokes equation \cite{Pi} we invoke the Messera-type principle to prove the existence, uniqueness of the AAP- mild solution for the Stokes equation in L p (?(TM))) space with p>d . We then establish the existence and uniqueness of the small AAP- mild solutions of the Navier-Stokes equation by using the fixed point argument. The asymptotic behaviour (exponential decay and stability) of these small solutions are also related. Our results extend also \cite{FaTa2013} for the forward asymptotic mild solution of the Navier-Stokes equation on the curved background.

Given a smooth manifold M (with or without boundary), in this paper we study the regularisation of traces for the global pseudo-differential calculus in the context of non-harmonic analysis. Indeed, using the global pseudo-differential calculus on manifolds (with or without boundary) developed in [30], the Calderón-Vaillancourt Theorem and the global functional calculus in [6], we determine the singularity orders in the regularisation of traces and the sharp regularity orders for the Dixmier traceability of the global Hörmander classes. Our analysis (free of coordinate systems) allows us to obtain non-harmonic analogues of several classical results arising from the microlocal analysis of regularised traces for pseudo-differential operators with symbols defined by localisations.

We consider the existence of a solution to the boundary value problem for the equation ?�div(D(?�u)??e ?�div(|?�u | p?? ?�u+ β 0 |?�u | ?? ?�u) )+au=f . This problem is derived from the mathematical modeling of crystal surfaces. The analytical difficulty is due to the fact that the smallest eigenvalue of the mobility matrix D(?�u) is not bounded away from 0 below and the inside operator is an exponential function composed with a linear combination of the p-Laplace operator and the 1-Laplace operator. Known existence results on problems related to ours either have to allow the possibility that the exponent in the equation be a measure or assume that data are suitably small in order to eliminate the possibility. In this paper we show the existence of a non-measure valued weak solution without any smallness assumption on the data. We achieve this by employing a power series expansion technique.