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In this paper we analyze the existence of solution to the homogeneous Dirichlet for a nonlinear elliptic singular problem, $$\begin{align*} \begin{cases} -\Delta u=\dfrac{|\nabla u|^q}{|u|^\alpha}+\lambda f & {\rm in}\ \Omega, \\ u=0 & {\rm on}\ \partial\Omega,\\ \end{cases} \end{align*} $$ { − Δ u = | ∇ u | q | u | α + λf in Ω , u = 0 on ∂Ω , where $ N\ge 2 $ N ≥ 2 , $ \Omega \subset \mathbb {R}^N $ Ω ⊂ R N is a bounded regular domain, q>2, $ \alpha α < q 2 , $ \lambda >0 $ λ > 0 and f is a nonnegative function belonging to a suitable Lebesgue space. Under additional hypotheses on the data, we are also able to show the uniqueness of the 'good' solution. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 and Ω be a bounded Lipschitz domain of R n . Assume that L R is a second-order divergence form elliptic operator having real-valued, bounded, symmetric, and measurable coefficients on L 2 (Ω) with the Robin boundary condition. In this article, via first obtaining the Hölder estimate of the heat kernels of L R , the authors establish a new atomic characterization of the Hardy space H L R p (Ω) associated with L R . Using this, the authors further show that, for any given p ∈ (n n + δ 0 , 1 ] , H z p (Ω) + L ∞ (Ω) = H L N p (Ω) = H L R p (Ω) ⫋ H L D p (Ω) = H r p (Ω) , where H L D p (Ω) and H L N p (Ω) denote the Hardy spaces on Ω associated with the corresponding elliptic operators respectively having the Dirichlet and the Neumann boundary conditions, H z p (Ω) and H r p (Ω) respectively denote the "supported type" and the "restricted type" Hardy spaces on Ω , and δ 0 ∈ (0 , 1 ] is the critical index depending on the operators L D , L N , and L R . The authors then obtain the boundedness of the Riesz transform ∇ L R - 1 / 2 on the Lebesgue space L p (Ω) when p ∈ (1 , ∞) [if p > 2 , some extra assumptions are needed] and its boundedness from H L R p (Ω) to L p (Ω) when p ∈ (0 , 1 ] or to H r p (Ω) when p ∈ (n n + 1 , 1 ] . As applications, the authors further obtain the global regularity estimates, in L p (Ω) when p ∈ (0 , p 0) and in H r p (Ω) when p ∈ (n n + 1 , 1 ] , for the inhomogeneous Robin problem of L R on Ω , where p 0 ∈ (2 , ∞) is a constant depending only on n, Ω , and the operator L R . The main novelties of these results are that the range (0 , p 0) of p for the global regularity estimates in the scale of L p (Ω) is sharp and that, in some sense, the space X : = H L R 1 (Ω) is also optimal to guarantee both the boundedness of ∇ L R - 1 / 2 from X to L 1 (Ω) or to H r 1 (Ω) and the global regularity estimate ‖ ∇ u ‖ L n n - 1 (Ω ; R n) ≤ C ‖ f ‖ X for inhomogeneous Robin problems with C being a positive constant independent of both u and f. [ABSTRACT FROM AUTHOR]

We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the first eigenvalue of the Laplacian on doubly connected domains when Robin and Dirichlet conditions are imposed on the outer and the inner part of the boundary, respectively. We provide that the spherical shell reaches the maximum of the first eigenvalue of this problem among a suitable class of domains when the measure, the outer perimeter and inner (n - 1) th quermassintegral are fixed. [ABSTRACT FROM AUTHOR]

We present a new method to determine the optimal regularity of positive solutions u ∈ C4(Ω\{0}) ∩ C0(Ω̄) of the Hénon-Hardy equation, i.e., (0.1) where Ω ⊂ ℝN (N ⩾ 4) is a bounded smooth domain with 0 ∈ Ω, α > −4, and p ∈ ℝ. It is clear that 0 is an isolated singular point of solutions of (0.1) and the optimal regularity of u in Ω relies on the parameter α. It is also important to see that the regularity of u at x = 0 determines the regularity of u in Ω. We first establish asymptotic expansions up to arbitrary orders at x = 0 of prescribed positive solutions u ∈ C4(Ω\{0}) ∩ C0(Ω̄) of (0.1). Then we show that the regularity at x = 0 of each positive solution u of (0.1) can be determined by some terms in asymptotic expansions of the related positive radial solution of the equation (0.1) with Ω = B, where B is the unit ball of ℝN. The main idea works for more general equations with singular weights. [ABSTRACT FROM AUTHOR]

We investigate a class of semipositone nonlinear elliptic boundary value problems in unbounded domains D of R n , n ≥ 3 . Under some mild-assumptions, we obtain the existence of a unique positive continuous solution and provide information about its global behavior. [ABSTRACT FROM AUTHOR]

In this article, we study some anisotropic singular perturbations for a class of linear elliptic problems. A uniform estimates for conforming $ Q_1 $ Q 1 finite element method are derived, and some other results of convergence and regularity for the continuous problem are proved. The arguments are based on the tensorization trick and standard elliptic regularity theory tools. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiple solutions of parametric quasi-linear systems of the gradient-type on the Sierpiński gasket arising in physical problems leading to nonlinear models involving reaction-diffusion equations, in problems on elastic fractal media or fluid flow through fractal regions. By using some critical point theorems, we give some new results to establish one, two and three weak solutions for our system. Finally, we give four examples to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper we focus on a certain class of anisotropic obstacle problems governed by a Leray–Lions operator. This problem is subject to homogeneous Neumann boundary conditions. By applying truncation techniques and the monotonicity method, we establish the existence of entropy solutions for the problem studied in the framework of anisotropic weighted Sobolev spaces with variable exponent. [ABSTRACT FROM AUTHOR]

We prove optimal Lipschitz regularity of solutions to Poisson's equation with measure data supported on a C 1 , Dini interface and with C 0 , Dini density. We achieve this by deriving pointwise gradient estimates on the interface, further showing the piecewise differentiability of solutions up to this surface. Our approach relies on perturbation arguments and estimates for the Green's function of the Laplacian. Additionally, we provide sharp counterexamples highlighting the minimality of our assumptions. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of a finite volume approximation of a convective diffusive elliptic problem with Neumann boundary conditions and L 1 data. To deal with the non-coercive character of the equation and the low regularity of the right hand-side we mix the finite volume tools and the renormalized techniques. To handle the Neumann boundary conditions we choose solutions having a null median and we prove a convergence result. We present also some numerical experiments in dimension 2 to illustrate the rate of convergence. [ABSTRACT FROM AUTHOR]

In this paper, we study the lower and upper bounds for Stokes eigenvalues by finite element schemes. For the schemes studied here, roughly speaking, the loss of the local approximation property of the discrete velocity and pressure spaces may lead to different computed bounds of the eigenvalues. Formally theoretical analysis is constructed based on certain mathematical hypotheses, and numerical experiments are given to illustrate the validity of the theory. [ABSTRACT FROM AUTHOR]

We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions. [ABSTRACT FROM AUTHOR]

In this paper, we prove some existence theorems for elliptic boundary value problems within the p(x)-Laplacian on a variable Sobolev space. For this purpose, the main problem is transformed into a fixed point problem and then fixed point arguments such as Schaefer’s and Schauder’s theorems are used. Our approach involves fewer stringent assumptions on the nonlinearity function than the prior findings. An interesting example is presented to examine the validity of the theoretical findings. [ABSTRACT FROM AUTHOR]

We study the resonant quasilinear problem - Δ p u = λ p u p - 1 + λ g (u) in Ω , u ≥ 0 in Ω , u | ∂ Ω = 0 , where Ω ⊂ R N is a smooth, bounded domain, λ p is the first eigenvalue of - Δ p in Ω , and g : [ 0 , + ∞) → R is a continuous and subcritical term. By means of variational arguments, we prove the existence of non-negative solutions for any λ > 0 ; positive solutions for sufficiently small λ > 0 . Our results generalize the ones recently obtained by different techniques in the case p = 2 . [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish an Ambrosetti-Prodi type result for the problem where is a parameter, is continuous and satisfies is continuous. with. [ABSTRACT FROM AUTHOR]

In this paper we prove existence of nonnegative bounded solutions for the non-autonomous prescribed mean curvature problem in non-parametric form on an open bounded domain Ω of R N . The mean curvature, that depends on the location of the solution u itself, is asked to be of the form f(x)h(u), where f is a nonnegative function in L N , ∞ (Ω) and h : R + ↦ R + is merely continuous and possibly unbounded near zero. As a preparatory tool for our analysis we propose a purely PDE approach to the prescribed mean curvature problem not depending on the solution, i.e. h ≡ 1 . This part, which has its own independent interest, aims to represent a modern and up-to-date account on the subject. Uniqueness is also handled in presence of a decreasing nonlinearity. The sharpness of the results is highlighted by mean of explicit examples. [ABSTRACT FROM AUTHOR]

In this article we consider the Dirichlet problem on a bounded domain Ω ⊂ R d with respect to a second-order elliptic differential operator in divergence form. We do not assume a divergence condition as in the pioneering work by Stampacchia, but merely assume that 0 is not a Dirichlet eigenvalue. The purpose of this article is to define and investigate a solution of the Dirichlet problem, which we call Perron solution, in a setting where no maximum principle is available. We characterise this solution in different ways: by approximating the domain by smooth domains from the interior, by variational properties, by the pointwise boundary behaviour at regular boundary points and by using the approximative trace. We also investigate for which boundary data the Perron solution has finite energy. Finally we show that the Perron solution is obtained as an H 0 1 -perturbation of a continuous function on Ω ¯ . This is new even for the Laplacian and solves an open problem. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R n + 1 , n ≥ 2 , be an open set with Ahlfors–David regular boundary satisfying the corkscrew condition. When Ω is quantitatively connected (i.e., it satisfies the Harnack chain condition) it is known that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures (i.e., its membership to the class A ∞ ) is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting, it is also known that these properties are stable under Fefferman–Kenig–Pipher's perturbations. Nonetheless, it has been an open problem to show whether, without connectivity, the previous two conditions remain equivalent or whether there is a Fefferman–Kenig–Pipher perturbation theory. In this paper we settle the question of whether, for general real elliptic operators in sets without any connectivity assumption, quantitative absolute continuity of the elliptic measure (expressed via a corona decomposition for the elliptic measure) and Carleson measure type estimates for bounded null solutions are equivalent. We show that any of these conditions is also equivalent to the fact that the Green function is comparable, in the corona sense, to the distance to the boundary. Our characterization has profound consequences. First, we extend Fefferman–Kenig–Pipher's perturbation result to sets which are not necessarily connected, and this is the first result in this general setting. Second, in the case of the Laplacian, and more generally for L 1 -Kenig–Pipher non-symmetric operators with variable coefficients, our conditions characterize the uniform rectifiability of the boundary. Last, our results generalize previous work in settings where quantitative connectivity is assumed. [ABSTRACT FROM AUTHOR]

We study the solvability of the second boundary value problem of the Lagrangian mean curvature equation arising from special Lagrangian geometry. By the parabolic method, we obtain the existence and uniqueness of the smooth uniformly convex solution, which generalizes the Brendle–Warren's theorem about minimal Lagrangian diffeomorphism in Euclidean metric space. [ABSTRACT FROM AUTHOR]

In this paper, we study coupled elliptic systems with gradient dependent right-hand sides and nonlinear boundary conditions, where the left-hand sides are driven by so-called double phase operators. By applying a surjectivity result for pseudomonotone operators along with an equivalent norm in the function space, we prove that the system has at least one nontrivial solution under very general assumptions on the data. Under slightly stronger conditions, we are also able to show that this solution is unique. As a result of independent interest, we further prove the boundedness of solutions to such elliptic systems by employing Moser’s iteration scheme.

Let Ω ⊂ R n be an open, bounded and Lipschitz set. We consider the Poisson problem for the p-Laplace operator associated to Ω with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison: we prove that the equality is achieved only if Ω is a ball and both the solution u and the right-hand side f of the Poisson equation are radial and decreasing. [ABSTRACT FROM AUTHOR]

We consider the linear elasticity system for a body coated with a thin elastic layer. The effect of the thin layer on that body can be described through an impedance operator linking the displacement and the traction on the interface. This operator cannot be in general explicitly computed for an arbitrary shape. In this paper, we consider the case of a linear elastic slab with Hooke's law. We compute by the Fourier transform the exact symbol of the impedance operator and prove that it is a pseudodifferential operator of order one. Then, we give a Padé-like approximation at order three (with respect to the thickness of the layer) for the impedance operator. The pattern of this approximation will be a helpful guide for approximating the impedance operator for a body whose boundary has nonzero curvature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p-Laplace operator, namely: λ F (β , Ω) = min ψ ∈ W 1 , p (Ω) \ { 0 } ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N - 1 ∫ Ω | ψ | p d x , where p ∈ ] 1 , + ∞ [ , Ω is a bounded, convex domain in R N , ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on R N. We show an upper bound for λ F (β , Ω) in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on the volume and the anisotropic perimeter of Ω , in the spirit of the classical estimates of Pólya (J Indian Math Soc (NS) 24:413–419, 1961) for the Euclidean Dirichlet Laplacian. We will also provide a lower bound for the torsional rigidity τ p (β , Ω) p - 1 = max ψ ∈ W 1 , p (Ω) \ { 0 } ∫ Ω | ψ | d x p ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N - 1 when β > 0. The obtained results are new also in the case of the classical Euclidean Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of strongly degenerate problems with critical exponential growth in R N , N ≥ 2 . We do not assume ellipticity condition on the operator and thus the maximum principle given by Lieberman (Commun Partial Differ Equ 16:311–361, 1991) can not be accessed. Therefore, a careful and delicate analysis is necessary and some ideas can not be applied in our scenario. The arguments developed in this paper are variational and our main result completes the study made in the current literature about the subject. Moreover, when N = 2 or N = 3 the solutions model the slow steady-state flow of a fluid of Prandtl-Eyring type. [ABSTRACT FROM AUTHOR]

In this paper, we consider a singular Adams' inequality with logarithmic weights in the unit ball of $ \mathbb {R}^4 $ R 4 . Our results extend the results of Zhu and Wang [Adams' inequality with logarithmic weights in $ \mathbb {R}^4 $ R 4 . Proc Amer Math Soc. 2021;149(8):3463–3472] on Adams' inequality with logarithmic weights to singular case. Then, we study the existence of solutions for some weighted mean field equations, relying on variational methods and the singular Adams' inequality with logarithmic weights we previously established. [ABSTRACT FROM AUTHOR]

In this paper, we consider inverse shape problems coming from diffuse optical tomography and the Helmholtz equation. In both problems, our goal is to reconstruct small volume interior regions from measured data on the exterior surface of an object. In order to achieve this, we will derive an asymptotic expansion of the reciprocity gap functional associated with each problem. The reciprocity gap functional takes in the measured Cauchy data on the exterior surface of the object. In diffuse optical tomography, we prove that a MUSIC-type algorithm that does not require evaluating the Green's function can be used to recover the unknown subregions. This gives an analytically rigorous and computationally simple method for recovering the small volume regions. For the problem coming from inverse scattering, we recover the subregions of interest via a direct sampling method. The direct sampling method presented here allows us to accurately recover the small volume region from one pair of Cauchy data, requiring less data than many direct sampling methods. We also prove that the direct sampling method is stable with respect to noisy data. Numerical examples will be presented for both cases in two dimensions where the measurement surface is the unit circle. [ABSTRACT FROM AUTHOR]

This paper studies local stability of solutions to a parametric multi-objective optimal control problem with constraints. By combining the scalarization method and non-scalarization methods, we show that if the unperturbed problem has a locally strong Pareto solution, then the Pareto solution set is locally Hölder continuous at a reference parameter. [ABSTRACT FROM AUTHOR]

With the help of De Giorgi's method, the embedding of the $ \mathfrak {B}_{m(x)}(\Omega,M,\gamma,\gamma _1,\delta,\frac {1}{q}) $ B m (x) (Ω , M , γ , γ 1 , δ , 1 q) class into $ C^{0,\alpha }(\Omega) $ C 0 , α (Ω) is proved. This result can be applied to the $ C^{0,\alpha } $ C 0 , α continuity to some elliptic equations with variable exponent. [ABSTRACT FROM AUTHOR]

In this paper, we consider the k-Hessian problem Sk(D2u) = b(x)f(u) in Ω, u = +∞ on ∂Ω, where Ω is a C∞-smooth bounded strictly (k − 1)-convex domain in R N with N ≥ 2, b ∈ C∞(Ω) is positive in Ω and may be singular or vanish on ∂Ω, f ∈ C[0, ∞) ∩ C1(0, ∞) (or f ∈ C 1 (R) ) is a positive and increasing function. We establish the first expansions (equalities) of k-convex solutions to the above problem when f is borderline regularly varying and Γ-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of f and principal curvatures of ∂Ω on the first expansion of solutions. For the latter, we find the principal curvatures of ∂Ω have no influences on the expansions. Our results and methods are quite different from the existing ones (including k = N). Moreover, we know the existence of k-convex solutions to the above problem (including k = N) is still an open problem when b possesses high singularity on ∂Ω and f satisfies Keller–Osserman type condition. For the radially symmetric case in the ball, we give a positive answer to this open problem, and then we further show the global estimates for all radial large solutions. [ABSTRACT FROM AUTHOR]

Let (X, d, μ) be a metric measure space satisfying the doubling condition and an L2-Poincaré inequality. Consider the nonnegative operator L generalized by a Dirichlet form on X. We will show that a solution u to (− ∂ t 2 + L) u = 0 on X × ℝ+ satisfies an α-Carleson condition if and only if u can be represented as the Poisson integral of the operator L with the trace in the generalized Morrey space L2,α(X), where α is a nonnegative function defined on a class of balls in X. This result extends the analogous characterization founded by R. Jiang, J. Xiao, D. Yang (2016) from the classical Morrey space on Euclidean space to the generalized Morrey space on the metric measure space. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Dirichlet problem in a bounded open set Ω ⊂ R n ( n ⩾ 2 ) for a class of second-order nonlinear elliptic equations with right-hand side f in L 1 (Ω) . We study the summability of entropy and weak solutions of this problem under the stronger assumption that f G (| f |) ∈ L 1 (Ω) , where G is a nonnegative increasing continuous function on [ 0 , + ∞) . We show how the summability of the solutions depends on the function G. Our conditions on G imply that L 1 + ε (Ω) ⊂ K G ⊂ L 1 (Ω) for every ε > 0 , where K G is the set of all measurable functions v on Ω such that v G (| v |) ∈ L 1 (Ω) . [ABSTRACT FROM AUTHOR]

In this paper, we study the following quasi-linear elliptic equation { − d i v (ϕ ( ∣ ∇ u ∣ ) ∇ u) = λ ψ ( ∣ u ∣ ) u + φ ( ∣ u ∣ ) u , i n Ω , u = 0 , o n ∂ Ω , where Ω ⊂ ℝN is a bounded domain, λ > 0 is a parameter. The function ψ(∣t∣)t is the subcritical term, and ϕ(∣t∣)t is the critical Orlicz-Sobolev growth term with respect to φ. Under appropriate conditions on φ, ψ and ϕ, we prove the existence of infinitely many weak solutions for quasi-linear elliptic equation, for λ ∈ (0, λ0), where λ0 > 0 is a fixed constant. [ABSTRACT FROM AUTHOR]

In this paper, we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface Σ with boundary ∂ Σ . Given a Riemannian metric g on Σ we consider functions of the form where σ i ≠ 0 for i = 1 , … , m , G g is the Green function of the Laplace-Beltrami operator on (Σ , g) with Neumann boundary conditions, R g is the corresponding Robin function, and h ∈ C 2 (Σ m , R) is arbitrary. We prove that for any Riemannian metric g, there exists a metric g ~ which is arbitrarily close to g and in the conformal class of g such that f g ~ is a Morse function. Furthermore we show that, if all σ i > 0 , then the set of Riemannian metrics for which f g is a Morse function is open and dense in the set of all Riemannian metrics. [ABSTRACT FROM AUTHOR]

In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function G behaves like a distance function to the boundary, in the sense that ∇ G (X) G (X) - ∇ D (X) D (X) 2 D (X) d X is the density of a Carleson measure, where D is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa's α -number of uniformly rectifiable sets. We believe that the method can be applied to many other problems at the intersection of PDE and geometric measure theory, and in particular, we are able to derive a generalization of the classical F. and M. Riesz theorem to the same class of elliptic operators as above. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded domain in R N with C 2 boundary and let K ⊂ ∂ Ω be either a C 2 submanifold of the boundary of codimension k < N or a point. In this article we study various problems related to the Schrödinger operator L μ = - Δ - μ d K - 2 where d K denotes the distance to K and μ ≤ k 2 / 4 . We establish parabolic boundary Harnack inequalities as well as related two-sided heat kernel and Green function estimates. We construct the associated Martin kernel and prove existence and uniqueness for the corresponding boundary value problem with data given by measures. To prove our results we introduce among other things a suitable notion of boundary trace. This trace is different from the one used by Marcus and Nguyen (Math Ann 374(1–2):361–394, 2019) thus allowing us to cover the whole range μ ≤ k 2 / 4 . [ABSTRACT FROM AUTHOR]

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections. [ABSTRACT FROM AUTHOR]

In this paper we study existence and uniqueness of solutions to Dirichlet problems as g (u) - div Du 1 + | D u | 2 = f in Ω , u = 0 on ∂ Ω , where Ω is an open bounded subset of R N ( N ≥ 2 ) with Lipschitz boundary, g : R → R is a continuous function and f belongs to some Lebesgue space. In particular, under suitable saturation and sign assumptions, we explore the regularizing effect given by the absorption term g(u) in order to get solutions for data f merely belonging to L 1 (Ω) and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in L N (Ω) as well as uniqueness if g is increasing. [ABSTRACT FROM AUTHOR]

For each k = 0 , ⋯ , n we construct a continuous phase f k , with f k (0) = (n - 2 k) π 2 , and viscosity sub- and supersolutions v k , u k , of the elliptic PDE ∑ i = 1 n arctan (λ i (H w)) = f k (x) such that v k - u k has an isolated maximum at the origin. It has been an open question whether the comparison principle would hold in this second order equation for arbitrary continuous phases f : R n ⊇ Ω → (- n π / 2 , n π / 2) . Our examples show it does not. [ABSTRACT FROM AUTHOR]

In the framework of Clifford analysis, we study higher order Dirac operators constructed with k-vectors. We find a necessary and sufficient condition to determine whether a function cancels them. [ABSTRACT FROM AUTHOR]

We study the behavior near the boundary corner point of weak bounded solutions to the oblique derivative problem for singular p-Laplacian equation with strong nonlinearities singular absorption term in a bounded plane sector. However, no work has been done for investigating the behavior of solutions to the oblique problem for singular p-Laplacian equation in a corner. [ABSTRACT FROM AUTHOR]

We present a simple yet accurate method to compute the adjoint double layer potential, which is used to solve the Neumann boundary value problem for Laplace's equation in three dimensions. An expansion in curvilinear coordinates leads us to modify the expression for the adjoint double layer so that the singularity is reduced when evaluating the integral on the surface. Then, to regularize the integral, we multiply the Green's function by a radial function with length parameter δ chosen so that the product is smooth. We show that a natural regularization has error O (δ 3) , and a simple modification improves the error to O (δ 5) . The integral is evaluated numerically without the need of special coordinates. We use this treatment of the adjoint double layer to solve the classical integral equation for the interior Neumann problem, altered to account for the solvability condition, and evaluate the solution on the boundary. Choosing δ = c h 4 / 5 , we find about O (h 4) convergence in our examples, where h is the spacing in a background grid. [ABSTRACT FROM AUTHOR]

This short letter considers the case of acoustic scattering by several obstacles in R d + r for r , d ≥ 1 of the form Ω × R r , where Ω is a smooth bounded domain in R d . As a main result, a von Neumann trace formula for the relative trace is obtained in this setting. As a special case, we obtain a dimensional reduction formula for the Casimir energy for the massive and massless scalar fields in this configuration Ω × R r per unit volume in R r . [ABSTRACT FROM AUTHOR]

We consider an eigenvalue problem for a double-phase differential operator with unbalanced growth. Using the Nehari method, we show that the problem has a continuous spectrum determined by the minimal eigenvalue of the weighted p-Laplacian. [ABSTRACT FROM AUTHOR]