Related papers: A Pexider equation containing the aritmetic mean
We prove that a Pfaffian system with coefficients in the critical space $L^2_\mathrm{loc}$ on a simply connected open subset of $\mathbb{R}^2$ has a non-trivial solution in $W^{1,2}_\mathrm{loc}$ if the coefficients are antisymmetric and…
We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H{\"o}lder continuous Lp valued functions…
We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $\phi (\Omega)$ parametrized by Lipschitz homeomorphisms $\phi $ defined on a fixed reference domain $\Omega$. Given two…
We study the existence, multiplicity and regularity results of non-negative solutions of following doubly nonlocal problem: $$ (P_\la) \left\{ \begin{array}{lr}\ds \quad (-\Delta)^{s_1}u+\ba (-\Delta)^{s_2}_{p}u = \la a(x)|u|^{q-2}u+…
In this paper, the connection between the functional inequalities $$ f\Big(\frac{x+y}{2}\Big)\leq\frac{f(x)+f(y)}{2}+\alpha_J(x-y) \qquad (x,y\in D)$$ and $$ \int_0^1f\big(tx+(1-t)y\big)\rho(t)dt \leq\lambda f(x)+(1-\lambda)f(y)…
We construct stable periodic solutions for a simple form nonlinear delay differential equation (DDE) with a periodic coefficient. The equation involves one underlying nonlinearity with the multiplicative periodic coefficient. The well-known…
We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced…
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,\mu_X)$ satisfying a $2$-Poincar\'e inequality. Given a…
The foundation of locally convex cone theory relies on order-theoretic concepts that induce specific topological frameworks. Within this structure, cones naturally possess three distinct topologies: lower, upper, and symmetric. In this…
We prove the local Lipschitz regularity of the local minimizers of scalar integral functionals of the form \begin{equation*} \mathcal{F}(v;\Omega)= \int_{\Omega} f (x, Dv) dx \end{equation*} under $(p,q)$-growth conditions. The main novelty…
We consider a hydrodynamic model of self-organized evolution of agents, with singular interaction kernel $\phi_\alpha(x)=1/|x|^{1+\alpha}$ ($0<\alpha<2$), in the presence of an additional external force. Well-posedness results are already…
A class of sufficient conditions of local regularity for suitable weak solutions to the nonstationary three-dimensional Navier-Stokes equations are discussed. The corresponding results are formulated in terms of functionals which are…
We consider a quasilinear parabolic stochastic partial differential equation driven by a multiplicative noise and study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine…
We prove the interior and global Lipschitz regularity results for a solution of fully nonlinear equations with $(p,q)$-growth. We prove that for a small gap $q-p$, a solution is locally or globally Lipschitz continuous. We also prove that a…
In this paper, we consider the following Klein-Gordon-Maxwell equations \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u+ V(x)u-(2\omega+\phi)\phi u=f(x,u)+h(x)&\mbox{in $\mathbb{R}^{3}$},\\ -\Delta \phi+ \phi u^2=-\omega u^2&\mbox{in…
In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak…
In this paper we are concerned with the initial boundary value problems of linear and semi-linear parabolic equations with mixed boundary conditions on non-cylindrical domains in spatial-temporal space. We obtain the existence of a weak…
We study the long-time asymptotics of solutions of the uniformly parabolic equation \[ u_t + F(D^2u) = 0 \quad {in} \R^n\times \R_+, \] for a positively homogeneous operator $F$, subject to the initial condition $u(x,0) = g(x)$, under the…
Let $\mathbf{X}$ be a class of metric spaces and let $\mathbf{P}_{\mathbf{X}}$ be the set of all $f:[0, \infty)\to [0, \infty)$ preserving $\mathbf{X},$ $(Y, f\circ\rho)\in\mathbf{X}$ whenever $(Y, \rho)\in\mathbf{X}.$ For arbitrary subset…
We study regularity for solutions of quasilinear elliptic equations of the form $\div \A(x,u,\nabla u) = \div \F $ in bounded domains in $\R^n$. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like…