Communications in Contemporary Mathematics

In this paper, we prove the global gradient estimates on the generalized Lebesgue spaces for weak solutions to elliptic quasilinear obstacle problems. It is worth noticing that the coefficients related to the obstacle problems are merely measurable with small BMO norms and the underlying domain does not satisfy any smoothness conditions.

In this paper, the existence and multiplicity of solutions for a quasilinear elliptic problem driven by the $\mathrm{\Phi }$-Laplacian operator is established. These solutions are also built as ground state solutions using the Nehari method. The main difficulty arises from the fact that the $\mathrm{\Phi }$-Laplacian operator is not homogeneous and the nonlinear term is indefinite.

In this paper, we prove the local existence and uniqueness of the solution with discontinuous density for the inhomogeneous non-resistive magnetohydrodynamics (MHD) equations on a $C^2$ bounded domain $\mathrm{\Omega }\subset R^3$ or $\mathrm{\Omega }=R^3$, if the initial data $({\rho }_0,u_0,B_0)\in L^{\infty }(\mathrm{\Omega })\times H^s(\mathrm{\Omega })\times W^{1,r}(\mathrm{\Omega })$ with $s>\frac{3}{2}-\frac{3}{r}>\frac{1}{2}$ satisfies $0<c_0\le {\rho }_0\le C_0<+\infty$.

We apply the difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous linear difference equation $y(x+2)+a(x)y(x+1)+b(x)y(x)=0,$ where the coefficients $a(x),b(x)\in \bar{\mathbb{Q}}(x)$ are rational functions in $x$ with coefficients in $\bar{\mathbb{Q}}$. We develop algorithms to compute the difference-differential Galois group associated to such an equation, and show how to deduce the differential-algebraic relations among the solutions from the defining equations of the Galois group.

This paper deals with an equivalent system to the nonlinear differential equation of Liénard type $(\phi (ẋ))\dot{}+f(x)\phi (ẋ)+g(x)=0$, where the range of the function $\phi$ is bounded. Sufficient conditions are obtained for the system to have at least one limit cycle. The proofs of our results are based on phase plane analysis of the system with the Poincaré–Bendixon theorem. Moreover, to show that these sufficient conditions are suitable in some sense, we also establish the results that the system has no limit cycles. Finally, some examples are given to illustrate our results.

We study the operator $-\mathrm{\text{div}}(A(x)\nabla \psi (x))$ restricted to a waveguide ${\mathrm{\Omega }}_{𝜀}\subset {ℝ}^3$, with heterogeneous function $A(x)$ constant in the longitudinal direction. The purpose is to obtain an effective operator, in the norm resolvent sense, when the diameter of ${\mathrm{\Omega }}_{𝜀}$ tends to zero with heterogeneity approaching a homogeneous situation (i.e. a constant function $A$). The effective operator presents a potential that, besides the traditional dependence on waveguide geometric properties, there is also a contribution from $A(x)$ which results, when combined with the curvature, for example, in the possibility of a repulsive interaction.

In this paper, we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of $f$-minimal hypersurfaces.

In this paper we prove that in a metric measure space $(X,d,𝔪)$ verifying the measure contraction property with parameters $K\in ℝ$ and $1<N<\infty$, any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to $𝔪$ and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map.

We establish the existence of solutions of fully nonlinear elliptic second-order equations like $H(v,Dv,D^{2}v,x)=0$ in smooth domains without requiring $H$ to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of $H$ at points at which $\left|D^{2}v\right|\le K$, where $K$ is any given constant. For large $\left|D^{2}v\right|$ some kind of relaxed convexity assumption with respect to $D^{2}v$ mixed with a VMO condition with respect to $x$ are still imposed. The solutions are sought in Sobolev classes. We also establish the solvability without almost any conditions on $H$, apart from ellipticity, but of a “cut-off” version of the equation $H(v,Dv,D^{2}v,x)=0$.

We establish sharp remainder terms of the $L^2$-Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups, yielding the inequalities with best constants. Our methods also give new sharp Caffarelli–Kohn–Nirenberg-type inequalities in ${ℝ}^n$ with arbitrary quasi-norms. We also present explicit examples to illustrate our results for different weights and in abelian cases.

We consider Auerbach bases in Banach spaces of dimension $n>2$. We show that there exist at least $(n-1)n/2+1$ such bases. This estimate follows from the calculation of the Lusternik–Schnirelmann category of the flag variety. A better estimate is obtained for generic smooth Banach spaces using Morse theory.

This paper is devoted to investigate the symmetry and monotonicity properties for positive solutions of fractional Laplacian equations. Especially, we consider the following fractional Laplacian equation with homogeneous Dirichlet condition: