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Using the Ekeland variational principle and the mountain pass lemma we prove some properties about potential homeomorphisms between a real Banach space and its dual. In particular, we show that a locally monotone homeomorphism is necessarily strictly monotone if it is potential.

Let $\Omega \subset \mathbb{R}^{n}$, $n>1$ and let $p,q\geq 2$. We consider the system of nonlinear Dirichlet problems \begin{equation*} \left\lbrace \begin{aligned} (Au)(x)& =N_{u}^{\prime }(x,u(x),v(x)), &&x\in\Omega, \\ -(Bv)(x)& =N_{v}^{\prime }(x,u(x),v(x)), &&x\in\Omega, \\ u(x) &= 0, &&x\in \partial \Omega,\\ v(x) &= 0, && x\in \partial \Omega, \end{aligned}\right. \end{equation*} where $N:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ is $C^1$ and is partially convex-concave and $A:W^{1,p}_0 (\Omega) \rightarrow(W^{1,p}_0 (\Omega) )^*$, $B: W^{1,p}_0 (\Omega) \rightarrow(W^{1,p}_0 (\Omega) )^*$ are monotone and potential operators. The solvability of this system is reached via the Ky–Fan minimax theorem.

The Robin problem for the diffusion equation in non-homogeneous media partial differential equation is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one applied by Chkadua, Mikhailov, Natroshvili. in [4]. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ Rn, n > 1 and let p,q ≥ 2. We consider the system of nonlinear Dirichlet problems... where N: R × R → R is C¹ and is partially convex-concave and A: W1,p0(Ω) →W-1,p0(Ω), B: W-1,q0 (Ω) → W-1,q0(Ω) are monotone and potential operators. The solvability of this system is reached via the Ky-Fan minimax theorem. [ABSTRACT FROM AUTHOR]

We show that certain vanishing properties defining closed subspaces of generalized Morrey spaces are preserved under the action of various classical operators of harmonic analysis, such as maximal operators, singular‐type operators, Hardy operators, and fractional integral operators. Those vanishing subspaces were recently used to deal with the delicate problem on the description of the closure of nice functions in Morrey norm. [ABSTRACT FROM AUTHOR]

We introduce the most general mixed fractional derivatives and integrals from three points of views: probability, the theory of operator semigroups, and the theory of generalized functions. The solutions to the resulting mixed fractional PDEs turned out to be representable in terms of of completely monotone functions in a certain class generalizing the usual Mittag-Leffler functions. [ABSTRACT FROM AUTHOR]

We prove that a generalized Fefferman-Phong type conditions on a pair of weights u and v is sufficient for the boundedness of the potential type operator from into . We also obtain an analogous estimates for their commutators with BMO symbols. We include some estimates for a generalized maximal operator in the variable context , and its fractional version, , between variable versions of type spaces, where and are exponents belonging to certain classes. [ABSTRACT FROM AUTHOR]

This paper concerns Hodge-Dirac operators D║ = d + δ acting in Lp(Ω, Λ) where Ω is a bounded open subset of Rn satisfying some kind of Lipschitz condition, Λ is the exterior algebra of Rn, d is the exterior derivative acting on the de Rham complex of differential forms on Ω, and δ is the interior derivative with tangential boundary conditions. In L²(Ω, Λ), δ = d* and D║ is self-adjoint, thus having bounded resolvents {(I + itD║) -1}t∈R as well as a bounded functional calculus in L²(Ω, Λ). We investigate the range of values pH < p < pH about p = 2 for which D║ has bounded resolvents and a bounded holomorphic functional calculus in Lp(Ω,Λ). On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which Lp(Ω, Λ) has a Hodge (or Helmholz) decomposition, being an open interval that includes 2. The Hodge-Laplacian Δ║ is the square of the Hodge-Dirac operator, i.e., -Δ║ = D║ 2, so it also has a bounded functional calculus in Lp(Ω, Λ) when pH < p < pH. But the Stokes operator with Hodge boundary conditions, which is the restriction of -Δ║ to the subspace of divergence free vector fields in Lp(Ω,Λ1) with tangential boundary conditions, has a bounded holomorphic functional calculus for further values of p, namely for max{1, pHS } < p < pH where pHS is the Sobolev exponent below pH, given by 1/pHS = 1/pH +1/n, so that pHS < 2n/(n+2). In 3 dimensions, pHS < 6/5. We show also that for bounded strongly Lipschitz domains O, pH < 2n/(n + 1) < 2n/(n - 1) < pH, in agreement with the known results that pH < 4/3 < 4 < pH in dimension 2, and pH < 3/2 < 3 < pH in dimension 3. In both dimensions 2 and 3, pHS < 1, implying that the Stokes operator has a bounded functional calculus in Lp(O, 1) when O is strongly Lipschitz and 1 < p < pH. [ABSTRACT FROM AUTHOR]

Abstract: We consider potential operators of order over sets in quasi-metric measure spaces and study their mapping properties from the subspace of functions in Hölder space vanishing on the boundary of , into the space , if . This is proved in a more general setting of generalized Hölder spaces with a given dominant of modulus of continuity. Statements of such a kind are known in the Euclidean case or in the case of quasimetric measure spaces with the cancellation property. In the general case, when the cancellation property fails, our proofs are based on a special treatment of the potential of a constant function, which in general has a regularity near the boundary of the type of the -th power of the distance to . An application to the case of spatial potentials over domains in and potentials over spherical caps is given. [Copyright &y& Elsevier]

We show that Riesz-type potential operators of order α over uniform domains Ω in ℝ map the subspace $H_{0}^{\lambda }(\Omega )$ of functions in Hölder space H(Ω) vanishing on ∂Ω, into the space H(Ω), if λ+ α≤1. This is proved in a more general setting of generalized Hölder spaces with a given dominant of continuity modulus. Statements of such a kind are known for instance for the whole space ℝ or more generally for metric measure spaces with cancellation property. In the case of domains in ℝ when the cancellation property fails, our proofs are based on a special treatment of potential of a constant function. [ABSTRACT FROM AUTHOR]

Abstract: Last years there was increasing an interest to the so-called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia''s extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vector-valued analogues. [Copyright &y& Elsevier]

We reexamine bifurcation questions for potential operators. Based on the Conley index theory we present a unified approach to bifurcations both at the origin and at infinity. Our results provide improvements of the existing theory. [ABSTRACT FROM AUTHOR]

By means of the limit and jump relations of classical potential theory with respect to the vectorial Helmholtz equation, a wavelet approach is established on a regular surface. The multiscale procedure is constructed in such a way that the emerging scalar, vectorial and tensorial potential kernels act as scaling functions. Corresponding wavelets are defined via a canonical refinement equation. A tree algorithm for fast decomposition of a tangential complex-valued vector field given on a regular surface is developed based on numerical integration rules. Some numerical test examples conclude the paper. [ABSTRACT FROM AUTHOR]

This paper represents a broadened version of the plenary lecture presented by the author at the conference Analytic Methods of Analysis and Differential Equations (AMADE-2003), September 4–9, 2003, Minsk, Belarus. We give a survey of investigations on ‘the variable exponent business’, concentrating mainly on recent advances in the operator theory and harmonic analysis in the generalized Lebesgue and Sobolev spaces L p (&CenterDot;) and W m , p (&CenterDot;) . [ABSTRACT FROM AUTHOR]

We generalize the Ap extrapolation theorem of Rubio de Francia to A∞ weights in the context of Muckenhoupt bases. Our result has several important features. First, it can be used to prove weak endpoint inequalities starting from strong-type inequalities, something which is impossible using the classical result. Second, it provides an alternative to the technique of good-λ inequalities for proving Lp norm inequalities relating operators. Third, it yields vector-valued inequalities without having to use the theory of Banach space valued operators. We give a number of applications to maximal functions, singular integrals, potential operators, commutators, multilinear Calderón–Zygmund operators, and multiparameter fractional integrals. In particular, we give new proofs, which completely avoid the good-λ inequalities, of Coifman''s inequality relating singular integrals and the maximal operator, of the Fefferman–Stein inequality relating the maximal operator and the sharp maximal operator, and the Muckenhoupt–Wheeden inequality relating the fractional integral operator and the fractional maximal operator. [Copyright &y& Elsevier]

We prove two-weight, weak type norm inequalities for potential operators and fractional integrals defined on spaces of homogeneous type. We show that the operators in question are bounded from Lp(v) to Lq,∞(u), 1, provided the pair of weights (u,v) verifies a Muckenhoupt condition with a “power-bump” on the weight u. [Copyright &y& Elsevier]

We extend some results of Amann about the topological degree for compact potential vector fields to potential operators of class (S)+. Using these results, we obtain the existence of a third buckled state of a thin elastic shell in R3 even in the case when the gradient of the potential energy of the shell might not be compact. [Copyright &y& Elsevier]

By means of the limit and jump relations of classical potential theory the framework of a wavelet approach on a regular surface is established. The properties of a multiresolution analysis are verified, and a tree algorithm for fast computation is developed based on numerical integration. As applications of the wavelet approach some numerical examples are presented, including the zoom-in property as well as the detection of high frequency perturbations. At the end we discuss a fast multiscale representation of the solution of (exterior) Dirichlet''s or Neumann''s boundary-value problem corresponding to regular surfaces. [Copyright &y& Elsevier]

The inverse problem of the calculus of variations for second-order nonlinear and linear systems of differential-difference equations is considered. The relationship between the formal potentiality of a linear system with constant coefficients and the parity of its characteristic function is established. [ABSTRACT FROM AUTHOR]

We study mixed norm inequalities for directional operators which appear applying the method of rotations to homogeneous operators with variable kernel and with the homogeneity of Riesz potentials. The results are sharp for a range of values of the parameter and for all its values when the inequalities are restricted to radial functions. [ABSTRACT FROM AUTHOR]