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Register a new userPotentials for $mathcal{A}$-quasiconvexity
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Authors
Bogdan Raita
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We show that each constant rank operator $mathcal{A}$ admits an exact potential $mathbb{B}$ in frequency space. We use this fact to show that the notion of $mathcal{A}$-quasiconvexity can be tested against compactly supported fields. We also show that $mathcal{A}$-free Young measures are generated by sequences $mathbb{B}u_j$, modulo shifts by the barycentre.
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We consider problems of static equilibrium in which the primary unknown is the stress field and the solutions maximize a complementary energy subject to equilibrium constraints. A necessary and sufficient condition for the sequential lower-semicontinuity of such functionals is symmetric ${rm div}$-quasiconvexity, a special case of Fonseca and Mullers $A$-quasiconvexity with $A = {rm div}$ acting on $R^{ntimes n}_{sym}$. We specifically consider the example of the static problem of plastic limit analysis and seek to characterize its relaxation in the non-standard case of a non-convex elastic domain. We show that the symmetric ${rm div}$-quasiconvex envelope of the elastic domain can be characterized explicitly for isotropic materials whose elastic domain depends on pressure $p$ and Mises effective shear stress $q$. The envelope then follows from a rank-$2$ hull construction in the $(p,q)$-plane. Remarkably, owing to the equilibrium constraint the relaxed elastic domain can still be strongly non-convex, which shows that convexity of the elastic domain is not a requirement for existence in plasticity.
We are concerned with the relaxation and existence theories of a general class of geometrical minimisation problems, with action integrals defined via differential forms over fibre bundles. We find natural algebraic and analytic conditions which give rise to a relaxation theory. Moreover, we propose the notion of ``Riemannian quasiconvexity for cost functions whose variables are differential forms on Riemannian manifolds, which extends the classical quasiconvexity condition in the Euclidean settings. The existence of minimisers under the Riemannian quasiconvexity condition has been established. This work may serve as a tentative generalisation of the framework developed in the recent paper: B. Dacorogna and W. Gangbo, Quasiconvexity and relaxation in optimal transportation of closed differential forms, textit{Arch. Ration. Mech. Anal.} (2019), to appear. DOI: texttt{https://doi.org/10.1007/s00205-019-01390-9}.
The soliton dynamics in the semiclassical limit for a weakly coupled nonlinear focusing Schrodinger systems in presence of a nonconstant potential is studied by taking as initial data some rescaled ground state solutions of an associate elliptic system.
In this paper, we consider the strongly coupled nonlinear Kirchhoff-type system with vanshing potentials: [begin{cases} -left(a_1+b_1int_{mathbb{R}^3}| abla u|^2dxright)Delta u+lambda V(x)u=frac{alpha}{alpha+beta}|u|^{alpha-2}u|v|^{beta},&xinmathbb{R}^3, -left(a_2+b_2int_{mathbb{R}^3}| abla v|^2dxright)Delta v+lambda W(x)v=frac{beta}{alpha+beta}|u|^{alpha}|v|^{beta-2}v,&xinmathbb{R}^3, u,vin mathcal{D}^{1,2}(mathbb{R}^3), end{cases}] where $a_i>0$ are constants, $lambda,b_i>0$ are parameters for $i=1,2$, $alpha,beta>1$ satisfy $alpha+betale4$, the nonlinear term $F(x,u,v)=|u|^alpha|v|^beta$ is not 4-superlinear at infinity, $V(x)$, $W(x)$ are nonnegative continuous potentials. By establishing some new estimates and truncation technique, we obtain the existence of positive vector solutions for the above system when $b_1+b_2$ small and $lambda$ large. Moreover, the asymptotic behavior of these vector solutions is also explored as $textbf{b}=(b_1,b_2)to bf{0}$ and $lambdatoinfty$. In particular, our results extend some known ones in previous papers that only deals with the case where $alpha,beta>2$ with $alpha+beta<6$.
We give the semiclassical asymptotic of barrier-top resonances for Schr{o}dinger operators on ${mathbb R}^{n}$, $n geq 1$, whose potential is $C^{infty}$ everywhere and analytic at infinity. In the globally analytic setting, this has already been obtained. Our proof is based on a propagation of singularities theorem at a hyperbolic fixed point that we establish here. This last result refines a theorem of the same authors, and its proof follows another approach.
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