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This paper is concerned with the fractional coupled Schrodinger system. By using the Nehari manifold and fibering map, we obtain the multiplicity and concentration of solutions for the given problem with steep potential wells, where some new estimates will be established. In particular, although there exist concave-convex nonlinearities in the coupled system, it is not necessary to assume that the corresponding Lebesgue norms of the weight functions of the convex terms need to be small enough.

In this paper we study the Dirichlet problem of a class of nonlinear elliptic equations which includes Monge–Ampere equation, K-Hessian equation and the usual linear elliptic equation. This problem becomes singular near the boundary of the domain. By carefully constructing sub-solutions we obtain the boundary Holder estimate for the convex solution to the problem when the domain satisfies exterior sphere condition.

We devote to investigating the following nonlinear reaction diffusion equations with nonlocal boundary conditions { u t = ∇ ⋅ ( ρ ( u ) ∇ u ) + k 1 ( t ) f ( u ) in D × ( 0 , t ⁎ ) , ∂ u ∂ ν = k 2 ( t ) ∫ D g ( u ) d x on ∂ D × ( 0 , t ⁎ ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 in D ‾ , where D is a bounded convex region in R n ( n ≥ 2 ) , and the boundary ∂D is smooth. By constructing some auxiliary functions and using differential inequality technique, we derive that the solution blows up at some finite time. Moreover, upper and lower bounds of the blow-up time are obtained.

We consider a multivalued nonlinear Duffing system driven by a nonlinear nonhomogeneous differential operator. We prove existence theorems for both the convex and nonconvex problems (according to whether the multivalued perturbation is convex valued or not). Also, we show that the solutions of the nonconvex problem are dense in those of the convex (relaxation theorem). Our work extends the recent one by Kalita-Kowalski (JMAA, https://doi.org/10.1016/j.jmaa. 2018.01.067).

We obtain a differential Harnack inequality for anisotropic curvature flow of convex hypersurfaces in Euclidean space with its speed given by a curvature function of homogeneity degree one in a certain class, and restrictions depending only on the initial data and the anisotropic factor which reflects the influence of the ambient space. Moreover, the pinching estimate for such flows is derived from the maximum principle for tensors.

In this paper, we investigate the motion of supersonic flow governed by 2-D steady isentropic irrotational relativistic Euler system entered into a convex duct. We will give the propagation and interactions of rarefaction waves between the upward and downward wall and obtain the global existence of piecewise continuous solutions. Furthermore, in some cases the vacuum state may appear and we also give a sufficient condition to the appearance of the vacuum in finite domain.

This paper is devoted to a class of important and general nonlocal fourth order elliptic problem Δ 2 u − ( 1 + λ ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) in R 3 , where Δ 2 = Δ ( Δ ) is the bi-harmonic operator, λ ≥ 0 is a constant. We focus on the case that f ( x , u ) involves a combination of convex and concave terms and the potential V ( x ) is allowed to be sign-changing. By new techniques, multiplicity results of two different type of solutions are established. Our results improves and generalizes that obtained in the literature.

A technique based on duality to obtain H 1 or other Sobolev regularity results for solutions of convex variational problems is presented. This technique, first developed in order to study the regularity of the pressure in the variational formulation of the Incompressible Euler equation, has been recently re-employed in Mean Field Games. Here, it is shown how to apply it to classical problems in relation with degenerate elliptic PDEs of p-Laplace type. This allows to recover many classical results via a different point of view, and to have inspiration for new ones. The applications include, among others, variational models for traffic congestion and more general minimization problems under divergence constraints, but the most interesting results are obtained in dynamical problems such as Mean Field Games with density constraints or density penalizations.

We prove a comparison principle for unbounded weak sub/super solutions of the equation λ u − div ( A ( x ) D u ) = H ( x , D u ) in Ω where A ( x ) is a bounded coercive matrix with measurable ingredients, λ ≥ 0 and ξ ↦ H ( x , ξ ) has a super linear growth and is convex at infinity. We improve earlier results where the convexity of H ( x , ⋅ ) was required to hold globally.

Brunn–Minkowski's theorem says that vol ( ( 1 − λ ) K + λ L ) 1 / n , for K, L convex bodies, is a concave function in λ, and assuming a common hyperplane projection of K and L, it is known that the volume itself is concave. The ‘a priori’ natural hypothesis of a common ( n − k )-plane projection of the sets turned out in the end not to imply the ( 1 / k )-th concavity of the volume function. In this paper we show which is the, somehow, best projection type assumption that is needed in order to get concavity for vol ( ( 1 − λ ) K + λ L ) 1 / k , characterizing also the equality case in the corresponding inequality. Moreover, we consider the same problem for its functional analogue: the Prekopa–Leindler inequality.

In this paper, by Karamata regular variation theory and constructing super and subsolutions, we obtain the asymptotic boundary behavior of convex solutions of Monge–Ampere equations with singular righthand sides.

The stochastic maximum principle for general controlled systems driven by fractional Brownian motions with Hurst parameter H ∈ ( 1 / 2 , 1 ) is obtained. The stochastic integrals with respect to fractional Brownian motions are Stratonovich-type integrals. The control domain need not be convex, and the diffusion coefficient can contain a control variable.

We establish a local Lipschitz regularity near the boundary of weak solutions to a general class of homogeneous quasilinear elliptic equations with Neumann boundary condition in bounded convex domains.

In this paper, for any symplectic matrix P, the existence of subharmonic P l -solutions of the first order non-autonomous superquadratic Hamiltonian systems is considered. Under the convex condition, the existence of infinitely many geometrically distinct P l -solutions is proved.

The goal of this paper is to clarify when a closed convex cone is invariant for a stochastic partial differential equation (SPDE) driven by a Wiener process and a Poisson random measure, and to provide conditions on the parameters of the SPDE, which are necessary and sufficient., Comment: 41 pages, 2 figures

This paper deals with a sharp smoothing effect for entropy solutions of one-dimensional scalar conservation laws with a degenerate convex flux. We briefly explain why degenerate fluxes are related with the optimal smoothing effect conjectured by Lions, Perthame, Tadmor for entropy solutions of multidimensional conservation laws. It turns out that generalized spaces of bounded variation B V Φ are particularly suitable –better than Sobolev spaces– to quantify the regularizing effect and to obtain traces as in BV. The function Φ in question is linked to the degeneracy of the flux. Up to the present, the Lax–Oleĭnik formula has provided optimal results for a uniformly convex flux. This formula is validated in this paper for the more general class of C 1 strictly convex fluxes –which contains degenerate convex fluxes– and enables the B V Φ smoothing effect in this class. We give a complete proof that for a C 1 strictly convex flux the Lax–Oleĭnik formula provides the unique entropy solution, namely the Kruzhkov solution.

We provide a-priori L ∞ bounds for classical positive solutions of semilinear elliptic systems in bounded convex domains when the nonlinearities are below the power functions v p and u q for any ( p , q ) lying on the critical Sobolev hyperbola. Our proof combines moving planes method and Rellich–Pohozaev type identities for systems. Our analysis widens the known ranges of nonlinearities for which classical positive solutions of semilinear elliptic systems are a priori bounded. Using these a priori bounds, and local and global bifurcation techniques, we prove the existence of positive solutions for a corresponding parametrized semilinear elliptic system.

In this paper, we establish the classical Hardy inequality in the solid torus and some variants of it. The general idea is to use the fact that Sobolev embeddings can be improved in the presence of symmetries. In all cases, using techniques that exploit the symmetry presented by the solid torus, we calculate the displayed best constants and we prove that they are the same as the standard Hardy best constants which appear in convex domains although the solid torus is not convex.

An evolution inclusion is considered in a separable Hilbert space. The right-hand side of the inclusion contains the subdifferential of a time-dependent proper convex lower semicontinuous function and a multivalued perturbation with nonempty closed, not necessarily, bounded values. Along with this inclusion we consider the inclusion with the perturbation term being convexified. We prove the existence of solutions and a density of the solution set of the original inclusion in a closure of the solution set of the inclusion with the convexified perturbation. In contrast to the known results of this kind we do not suppose that the convex function has the compactness property and that the values of the perturbation are bounded sets. An example of a perturbation with closed unbounded values satisfying the conditions of the main theorems is given.

We estimate the Jacobian of harmonic mapping of the unit disk onto a smooth and convex Jordan domain via the boundary function and the boundary curvature of the image domain. By using this result we make some asymptotically sharp estimates of constant of quasiconformality for harmonic diffeomorphisms between the unit disk and the convex domains via their boundary mappings.

For each quasi-metric space X we consider the convex lattice SLip 1 ( X ) of all semi-Lipschitz functions on X with semi-Lipschitz constant not greater than 1. If X and Y are two complete quasi-metric spaces, we prove that every convex lattice isomorphism T from SLip 1 ( Y ) onto SLip 1 ( X ) can be written in the form T f = c ⋅ ( f ∘ τ ) + ϕ , where τ is an isometry, c > 0 and ϕ ∈ SLip 1 ( X ) . As a consequence, we obtain that two complete quasi-metric spaces are almost isometric if, and only if, there exists an almost-unital convex lattice isomorphism between SLip 1 ( X ) and SLip 1 ( Y ) .

We consider two problems on sections of convex bodies in hyperbolic space. The first one is a modified version of the Busemann–Petty problem. We look at conditions that guarantee a positive answer to this problem in all dimensions. The second problem is an analogue of a result of Makai, Martini, and Odor about origin-symmetry. If in every direction the parallel section function has a critical value at zero, then the body is origin-symmetric. For both problems we use Fourier transform techniques.

In this paper we prove that compactness can be characterized by means of the existence of a fixed point for some classes of mappings defined on convex closed subsets of the space l 1 . Nominally, our result involves nonexpansive mappings, uniformly Lipschitzian mappings and cascading nonexpansive mappings. We also extend the results to some more general classes of Banach spaces.

We prove a Liouville type theorem for stable solutions of the equation −div(|Du|p−2Du)=f(u) in the whole space RN, where f is a non-negative, non-decreasing and convex function. We apply that result to obtain another Liouville type theorem for non-negative solutions of similar equations in the half-space R+N.

We study the conormal derivative problem for an elliptic equation of p-Laplacian type with discontinuous coefficients in a non-smooth domain in order to look for the minimal assumptions necessary to have the nonlinear Calderon–Zygmund theory for such problem. Under the assumptions that the nonlinear operator is sufficiently close to the p-Laplacian operator in BMO semi-norm and the boundary of the domain can be locally approximated by the convex boundary, we prove that both the gradient and the associated nonhomogeneous term belong to the same L q space for every q ∈ [ p , ∞ ) . As far as the domain is concerned, our regularity assumption on the boundary is weaker than any other one reported in this direction.

Using the convex process theory and the majorizing technique, we study the convergence issues of the iterative sequences generated by the extended Newton method for solving the conic inequality system F ≥ C 0 defined by a cone C and a Frechet differentiable function F satisfying the (extended) weak γ-condition. Convergence criterion of the extended Newton method is presented. As an application, we establish, when F is analytic, an extended α-theory similar to Smale's α-theory for nonlinear analytic equations.

In this paper we study this chemotaxis-system { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + g ( u ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , defined in a convex smooth and bounded domain Ω of R n , with n ≥ 1 , and endowed with homogeneous Neumann boundary conditions, being χ > 0 and g a sort of logistic function obeying growth technical restrictions. Specifically, once an appropriate definition of very weak solution is given, our main result deals with the global existence of such solutions for any nonnegative initial data ( u 0 , v 0 ) ∈ C 0 ( Ω ¯ ) × C 2 ( Ω ¯ ) , and under zero-flux boundary condition on v 0 .

In this paper, we establish the demiclosedness principle for the class of mean nonexpansive mappings, introduced in 2007 by Goebel and Japon Pineda, defined on closed, convex subsets of Banach spaces satisfying Opial's condition. We also establish the demiclosedness principle for a subclass of the mean nonexpansive maps whose domains are closed, bounded, convex sets in uniformly convex spaces. These results extend known demiclosedness results for nonexpansive maps and lead to some fixed point results for mean nonexpansive maps which partially answer an open question posed by Goebel and Japon Pineda.

We analyze existence and uniqueness of l 2 -solutions of the implicit discrete Nagumo reaction–diffusion equation. We study the infinite-dimensional problem variationally and describe corresponding potentials which have either the convex or mountain pass geometry. Consequently, we show that the implicit Nagumo equation has a solution for all reaction parameters λ ∈ R , at least for small time discretization steps h. Moreover, the solution is unique in the bistable case, λ > 0 .

In this note, we study a semilinear system involving the curl operator in a bounded and convex domain in R 3 , which is a limiting form of the Ginzburg–Landau model for superconductors in three dimensions for a large value of the Ginzburg–Landau parameter. We show the existence and establish the L ∞ estimate associated with the boundary data for the weak solution of this system. As an application, our result gives a lower bound of the strength of the applied field such that the superconducting state is locally stable.

We establish an algebraic contraction rate in a modified Wasserstein distance for solutions of scalar conservation laws with uniformly convex flux. We also show that our estimate is optimal w.r.t. scaling in time and discuss why it gives non-trivial information in relation to the stability of the rarefaction wave.

We investigate the asymptotic behavior, as t goes to infinity, for a semilinear hyperbolic equation with asymptotically smal dissipation and convex potential. We prove that if the damping term behaves like K/t^\alpha for t large enough, k>0 and 0, Comment: 7 pages

In this study, we consider some equations that admit compacton solutions induced by a non-convex convection. Some conservation laws are derived for the nonlinearly self-adjoint equations based on differential substitutions. We also derive some conservation laws using the multipliers method.

We present a new approach to studying a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is concave (i.e., ( p − 1 )-sublinear) near zero and convex (i.e., ( p − 1 )-superlinear) near ±∞. The reaction term is not assumed to be odd. We show that for all small values of the parameter λ > 0 , the problem has infinitely many nodal solutions.

We establish global Holder estimates for solutions to inhomogeneous linearized Monge–Ampere equations in two dimensions with the right hand side being the divergence of a bounded vector field. These equations arise in the semi-geostrophic equations in meteorology and in the approximation of convex functionals subject to a convexity constraint using fourth order Abreu type equations. Our estimates hold under natural assumptions on the domain, boundary data and Monge-Ampere measure being bounded away from zero and infinity. They are an affine invariant and degenerate version of global Holder estimates by Murthy-Stampacchia and Trudinger for second order elliptic equations in divergence form.

The proof of Proposition 1.1 in S. Hong et al. (2014) [3] is erroneous and we reprove it.

In this paper, we will construct examples of non-convex constant mean curvature spheres in R 3 with a local perturbation of the Euclidean metric. When the perturbation area becomes small and converges to the origin, the non-convex constant mean curvature sphere converges in the Hausdorff sense to the singular space of two tangent unit spheres.

In this paper we study the magnitude of the multivariate Christoffel functions on general starlike and convex domains. We will obtain sharp pointwise upper bounds which will reveal the behavior of Christoffel functions near the boundary of the domain and their dependence on the geometry of the domain.

Let K , E , L be convex bodies, dim ⁡ L ≤ 1 and K = L + E , a sausage. In this case vol ( λ K + ( 1 − λ ) E ) = λ vol ( K ) + ( 1 − λ ) vol ( E ) . We prove that under the sole assumption that K and E have an equal volume projection (or a common maximal volume section), if the above equality holds for just one value in ( 0 , 1 ) , then K = L + E with dim ⁡ L ≤ 1 . However, even having equality for all λ ∈ [ 0 , 1 ] , if no extra assumption on K , E is done, such a characterization is not possible. This problem is connected with a conjecture relating the roots of the Steiner polynomial of a pair of convex bodies to their relative inradius. Counterexamples for the general case are explicitly given. In the same line, a counterexample to a conjecture by Matheron on inner parallel bodies is also shown.

The class consisting of analytic functions f in the unit disk satisfying f + α z f ′ + γ z 2 f ″ subordinated to some function h is considered. The Bohr radius for this class is obtained when h is respectively convex or starlike. The Bohr radius for analytic functions mapping the unit disk into a concave-wedge domain as well as for bounded harmonic mappings are also established.

We consider solutions to a differential inclusion x ˙ ∈ F ( x ) constrained to a compact convex set Ω. Here F is a compact, possibly non-convex valued, Lipschitz continuous multifunction, whose convex closure co F satisfies a strict inward pointing condition at every boundary point x ∈ ∂ Ω . Given a reference trajectory x ⁎ ( . ) taking values in an e-neighborhood of Ω, we prove the existence of a second (approximating) trajectory x : [ 0 , T ] ↦ Ω which satisfies the linear estimate ‖ x ( . ) − x ⁎ ( . ) ‖ AC ( [ 0 , T ] ) ⩽ K e , if one of the following two cases occurs: (i) the initial datum x ( 0 ) = x 0 is given, but lies in a compact set containing only points where the boundary ∂Ω is smooth; (ii) the initial point x ( 0 ) ∈ Ω of the approximating trajectory x ( . ) can be chosen arbitrarily. Subsequently we employ these linear AC -estimates to establish conditions for normality of the generalized Euler–Lagrange condition for optimal control problems with state constraints, in which we have an integral term in the cost. We finally provide an illustrative example which underlines the fact that, if conditions (i) and (ii) above are not satisfied, then we can find a degenerate minimizer.

We prove sharp L p -Sobolev estimates for averaging operators associated with a certain type of convex hypersurface in R 3 .

In this paper we study the H 2 global regularity for solutions of the p ( x ) -Laplacian in two-dimensional convex domains with Dirichlet boundary conditions. Here p : Ω → [ p 1 , ∞ ) with p ∈ Lip ( Ω ¯ ) and p 1 > 1 .

In this paper, we study the evolution of convex hypersurfaces by inverse mean curvature minus an external force field c. We prove that the flow will preserve the convexity for any c. If c 1 H on the initial surface, we prove that the flow will expand the hypersurface for all time, and after rescaling the hypersurface will converge to a sphere. If c > 1 H on the initial surface, we show that the maximum existence time of the flow is finite, and the hypersurface will contract to a point when approaching the final time.

We investigate an overdetermined problem for the torsion equation Δ u = − 1 with two kinds of free boundary conditions in an unbounded domain and present a symmetry result in the case of 2D. The proof is based on the maximum principle. In the proof, we use the sliding method and some convex arguments.

In this paper, we use the Perron method to prove the existence of viscosity solutions to a class of Monge–Ampere equations on exterior domains in R n ( n ≥ 2 ) with prescribed asymptotic behavior at infinity. This extends the Caffarelli–Li’s result on det ( D 2 u ) = 1 . We also obtain the existence of entire convex viscosity solutions to the Monge–Ampere equations with prescribed asymptotic behavior at infinity.

The lateral boundary differentiability with respect to x is shown for solutions of parabolic differential equations in nondivergence form on cylindrical convex domains.

In this paper, we study supnorm and modified Holder estimates for the integral solution of the ∂ -equation on a class of convex domains of general type in C 2 that includes many infinite type examples.

We investigate the roots of Wills polynomials of convex bodies. We study the structure, showing that the set of roots in the upper half plane is a convex cone, monotonous with respect to the dimension. In particular, we give its precise description for dimensions n = 2 , 3 . We also show that for n ≤ 7 this cone is completely contained in the (open) left half plane, which is not true in dimensions ≥ 14 . Moreover, we study the size of the roots of the Wills polynomial, bounding them in terms of functionals like the in- and circumradius of the set. We also relate the roots of the Steiner and the Wills polynomials.

We study the existence of standing waves for a class of quasilinear Schrodinger equations − Δ u + V ( x ) u − [ Δ ( u 2 ) ] u = μ | u | q − 2 u + | u | r − 2 u , x ∈ R N , where N ≥ 3 , 2 q 4 , 4 r 2 ( 2 ∗ ) ≔ 4 N / ( N − 2 ) and μ ≥ 0 . V is a positive potential. One main difficulty in dealing with this problem seems to be that of obtaining the boundedness of a (PS) sequence for the corresponding functional. We overcome this difficulty by using Jeanjean’s result [L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer type problem, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 787–809]. Our results answer some questions raised in [J. M. do O, U. Severo, Quasilinear Schrodinger equations involving concave and convex nonlinearities, Comm. Pure Appl. Anal. 8 (2009) 621–644].