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The probe and singular sources methods are two well-known classical direct reconstruction methods in inverse obstacle problems governed by partial differential equations. In this paper, by considering an inverse obstacle problem governed by the Laplace equation in a bounded domain as a prototype case, an integrated theory of the probe and singular sources methods is proposed. The theory consists of three parts: (i) introducing the singular sources method combined with the notion of the probe method; (ii) finding a third indicator function whose two ways decomposition yields the indicator functions in the probe and singular sources methods; (iii) finding the completely integrated version of the probe and singular sources methods. [ABSTRACT FROM AUTHOR]

The article is devoted to the existence of solutions of a system of integro-differential equations involving the drift terms in the case of the normal diffusion and the influx/efflux terms proportional to the Dirac delta function. The proof of the existence of solutions is based on a fixed point technique. We use the solvability conditions for the non- Fredholm elliptic operators in unbounded domains. We emphasize that the study of the systems is more difficult than of the scalar case and requires to overcome more cumbersome technicalities. [ABSTRACT FROM AUTHOR]

The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function in the presence of the drift term. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains and discuss how the introduction of the transport term influences the regularity of the solutions. [ABSTRACT FROM AUTHOR]

We prove the existence in the sense of sequences of solutions for some system of integro-differential type equations in two dimensions containing the normal diffusion in one direction and the anomalous diffusion in the other direction in H 2 (R 2 , R N) using the fixed point technique. The system of elliptic equations contains second order differential operators without the Fredholm property. It is established that, under the reasonable technical assumptions, the convergence in L 1 (R 2) of the integral kernels yields the existence and convergence in H 2 (R 2 , R N) of the solutions. We emphasize that the study of the systems is more difficult than of the scalar case and requires to overcome more cumbersome technicalities. [ABSTRACT FROM AUTHOR]

We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is u ≡ 0 . We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions. [ABSTRACT FROM AUTHOR]

Petrecca and Röser (Mathematische Zeitschrift 291:395–419, 2018) and Schueth (Ann Global Anal Geom 52:187–200, 2017) had shown that for a generic G-invariant metric g on certain compact homogeneous spaces M = G / K (including symmetric spaces of rank 1 and some Lie groups), the spectrum of the Laplace-Beltrami operator Δ g was real G-simple. The same is not true for the complex version of Δ g when there is a presence of representations of complex or quaternionic type. We show that these types of representations induces a Q 8 -action that commutes with the Laplacian in such way that G-properties of the real version of the operator have to be understood as (Q 8 × G) -properties on its corresponding complex version. [ABSTRACT FROM AUTHOR]

In this paper, we prove Hopf's lemma for the Logarithmic Laplacian. At first, we introduce the strong minimum principle. Then Hopf's lemma for the Logarithmic Laplacian in the ball is proved. On this basis, Hopf's lemma of the Logarithmic Laplacian is extended to any open set with the property of the interior ball. Finally, an example is given to explain Hopf's lemma can be applied to the study of the symmetry of the positive solution of the nonlinear Logarithmic Laplacian problem by the moving plane method. [ABSTRACT FROM AUTHOR]

We consider the following Kirchhoff equation in the Sobolev critical case with combined power nonlinearities having prescribed mass ∫ R 3 | u | 2 = c 2 , where a , c , μ > 0 are positive constants, b > 0 is a positive parameter, 2 < q < p ¯ : = 2 + 8 3 which is L 2 -critical exponent. For the L 2 -subcritical case 2 < q < 10 3 and Sobolev critical case, Li et al. (2021) proved that (K) has a solution which is ground state solution and corresponds to local minima of the associated energy functional. Here we extend the result in Li et al. (2021) by proving that (K) has the second solution which is not a ground state and is located at a mountain-pass level of the energy functional. Meanwhile, let u b are normalized solutions of mountain-pass type to (K) , then u b → u in H 1 (R 3) as b → 0 up to a subsequence, where u ∈ H 1 (R 3) is a normalized solution of mountain-pass type to - a ▵ u = λ u + μ | u | q - 2 u + | u | 4 u in R 3. Our results also extend the results of Soave (J Differ Equ 269:6941–6987, 2020; J Funct Anal 279:108610, 2020). [ABSTRACT FROM AUTHOR]

We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension d. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber k with a number of degrees of freedom scaling as k d - 1 / 2 , which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as k d to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation. [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]

We consider propagation of two-dimensional waves on the infinite hexagonal (honeycomb) lattice. Namely, we study the discrete Helmholtz equation in hexagonal lattices without and with a boundary. It is shown that for some configurations these problems can be equivalently reduced to similar problems for the triangular lattice. Based on this fact, new results are obtained for the existence and uniqueness of the solution in the case of the real wave number k ∈ ( 0 , 6 ) ∖ { 2 , 3 , 2 } {k\in(0,\sqrt{6})\setminus\{\sqrt{2},\sqrt{3},2\}} for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number k ∈ ( 0 , 2 ) ∪ ( 2 , 6 ) {k\in(0,\sqrt{2})\cup(2,\sqrt{6})} for the exterior Dirichlet problem. [ABSTRACT FROM AUTHOR]

We study the relaxation problem for a two-fluid Euler-Poisson system. We prove the global-in-time convergence of the system for smooth solutions near the constant equilibrium states. The limit system is the two-fluid drift-diffusion system as the relaxation time tends to zero. In the proof, we establish uniform energy estimates of smooth solutions for all the parameters and the time. These estimates allow us to pass to the limit in the system to obtain the limit system. Moreover, the global convergence rate of the solutions is obtained by stream function techniques. [ABSTRACT FROM AUTHOR]

We revisit the well-known Brezis-Nirenberg problem - Δ u = u N + 2 N - 2 + ε u , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω , where ε > 0 and Ω ⊂ R N are a smooth bounded domain with N ≥ 3 . The existence of multi-bump solutions to above problem for small parameter ε > 0 was obtained by Musso and Pistoia (Indiana Univ Math J 51:541–579, 2002). However, to our knowledge, whether the multi-bump solutions are non-degenerate that is open. Here, we give some straightforward answer on this question under some suitable assumptions for the Green's function of - Δ in Ω , which enriches the qualitative analysis on the solutions of Brezis-Nirenberg problem and can be viewed as a generalization of Grossi (Nonlinear Differ Equ Appl 12:227–241, 2005) where the non-degeneracy of a single-bump solution has been proved. And the main idea is the blow-up analysis based on the local Pohozaev identities. [ABSTRACT FROM AUTHOR]

We present straightforward conditions which ensure that a strongly elliptic linear operator L generates an analytic semigroup on Hölder spaces on an arbitrary complete manifold of bounded geometry. This is done by establishing the equivalent property that L is ‘sectorial,’ a condition that specifies the decay of the resolvent (λ I - L) - 1 as λ diverges from the Hölder spectrum of L. A key step is that we prove existence of this resolvent if λ is sufficiently large using a geometric microlocal version of the semiclassical pseudodifferential calculus. The properties of L and e - t L we obtain can then be used to prove well-posedness of a wide class of nonlinear flows. We illustrate this by proving well-posedness on Hölder spaces of the flow associated with the ambient obstruction tensor on complete manifolds of bounded geometry. This new result for a higher-order flow on a noncompact manifold exhibits the broader applicability of our technique. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel second-order generalised rotational discrete gradient scheme for numerically approximating the orthonormal frame gradient flow of biaxial nematic liquid crystals. This scheme relies on reformulating the original gradient flow system into an equivalent generalised “rotational” form. A second-order discrete gradient approximation of the energy variation is then devised such that it satisfies an energy difference relation. The proposed numerical scheme has two remarkable properties: (i) it strictly obeys the orthonormal property of the tensor field and (ii) it satisfies the energy dissipation law at the discrete level, regardless of the time step sizes. We provide ample numerical results to validate the accuracy, efficiency, unconditional stability and SO(3)-preserving property of this scheme. In addition, comparisons of the simulation results between the biaxial orthonormal frame gradient flow model and uniaxial Oseen–Frank gradient flow are made to demonstrate the ability of the former to characterize non-axisymmetric local anisotropy. [ABSTRACT FROM AUTHOR]

The zero-relaxation limit of Euler–Poisson systems for ion dynamics in a slow time scale leads to drift-diffusion equations. This fact was justified in previous works. This paper concerns global error estimates between solutions of Euler–Poisson systems and that of drift-diffusion equations. In the proof, we use Sobolev inequalities and employ a multiplier related to the electric potential. [ABSTRACT FROM AUTHOR]

We investigate the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs in R d where the edge cost between two points is given by a pth power of their Euclidean distance. This includes e.g. the travelling salesperson problem and the bounded degree minimum spanning tree. We establish in particular almost sure convergence, as n grows, of a suitable renormalization of the random minimum cost, if the points are uniformly distributed and d ≥ 3 , 1 ≤ p < d . Previous results were limited to the range p < d / 2 . Our proofs are based on subadditivity methods and build upon new bounds for random instances of the Euclidean bipartite matching problem, obtained through its optimal transport relaxation and functional analytic techniques. [ABSTRACT FROM AUTHOR]

The work is motivated by the Faraday cage effect. We consider the Helmholtz equation over a 3D domain containing a thin heterogeneous interface of thickness $ \delta \ll 1 $ δ ≪ 1. The layer has a δ-periodic structure in the in-plane directions and is cylindrical in the third direction. The periodic layer has one connected component and a collection of isolated regions. The isolated region in the thin layer represents air or liquid, and the connected component represents a solid metal grid with a δ thickness. The main issue is created by the contrast of the coefficients in the air and in the grid and that the zero-order term has a complex-valued coefficient in the connected faze while a real-valued in the complement. An asymptotic analysis with respect to $ \delta \to 0 $ δ → 0 is provided, and the limit Helmholtz problem is obtained with the Dirichlet condition on the interface. The periodic unfolding method is used to find the limit. [ABSTRACT FROM AUTHOR]

The method of fundamental solutions has been mainly applied to wave scattering problems in bounded domains and to our knowledge there have not been works addressing density results for general shapes, or addressing the calculation of the complex resonance frequencies that occur in exterior problems. We prove density and convergence of the fundamental solutions approximation in the context of wave scattering problems, with and without a priori knowledge of the frequency, which is of particular importance to detect resonance frequencies for trapping domains. We also present several numerical results that illustrate the good performance of the method in the calculation of complex resonance frequencies for trapping and non trapping domains in 2D and 3D. [ABSTRACT FROM AUTHOR]

This study is focused on finding the solution to the initial value problem for the fractional elliptic equations driven by the Wiener process. First, with some strong conditions on the input data, we establish the regularity of the solution. With relaxed conditions of the input data function, our forward problem is ill-posed in the sense of Hadamard. For this ill-posed problem, the truncation method is used to construct a regularized solution. Under prior assumptions for the exact solution, the convergence rate is obtained. [ABSTRACT FROM AUTHOR]

It is established existence and multiplicity of solutions for semilinear elliptic problems defined in the whole space R N considering subcritical nonlinearities with some parameters. Here we emphasize that our nonlinearities can be sign-changing functions. The main difficulty is proving the existence of nontrivial solutions by using the Nehari method, taking into account that the Lagrange multipliers theorem cannot be directly applied in our setting. In fact, we consider the case where the fibering map admits inflection points. In other words, we consider the case where the Nehari set admits degenerate critical points. Hence our main contribution is to consider a huge class of semilinear elliptic problems where the standard Nehari method cannot be applied. Using some fine estimates and recovering some compactness results together with the nonlinear Rayleigh quotient, we prove that our main problem admits at least three nontrivial solutions depending on the parameters. [ABSTRACT FROM AUTHOR]

Here, considering - ∞ < a < N - p p , a ≤ e ≤ a + 1 , d = 1 + a - e and p ∗ : = p ∗ (a , e) = Np N - d p , the existence of positive solution of a weighted p-Laplace equation involving vanishing potentials - Δ ap u + V (x) | x | - e p ∗ | u | p - 2 u = | x | - e p ∗ f (u) in R N is proved, where the potential V can vanish at infinity with exponential decay and f is a function with subcritical growth of class C 1 . We use Del Pino & Felmer’s arguments to overcome the lack of compactness and the Moser iteration method with Caffarelli–Kohn–Nirenberg inequality to obtain estimates of the solution in L ∞ (R N). [ABSTRACT FROM AUTHOR]

Being driven by the goal of finding edge modes and of explaining the occurrence of edge modes in the case of time-modulated metamaterials in the high-contrast and subwavelength regime, we analyse the topological properties of Floquet normal forms of periodically parameterized time-periodic linear ordinary differential equations d dt X = A α (t) X α ∈ T d . In fact, our main goal being the question whether an analogous principle as the bulk-boundary correspondence of solid-state physics is possible in the case of Floquet metamaterials, i.e., subwavelength high-contrast time-modulated metamaterials. This paper is a first step in that direction. Since the bulk-boundary correspondence states that topological properties of the bulk materials characterize the occurrence of edge modes, we dedicate this paper to the topological analysis of subwavelength solutions in Floquet metamaterials. This work should thus be considered as a basis for further investigation on whether topological properties of the bulk materials are linked to the occurrence of edge modes. The subwavelength solutions being described by a periodically parameterized time-periodic linear ordinary differential equation d dt X = A α (t) X α ∈ T d , we put ourselves in the general setting of periodically parameterized time-periodic linear ordinary differential equations and introduce a way to (topologically) classify a Floquet normal form F, P of the associated fundamental solution X α (t) = P (α , t) exp (t F α) α ∈ T d . This is achieved by analysing the topological properties of the eigenvalues and eigenvectors of the monodromy matrix X α (T) and the Lyapunov transformation P (α , t) . The corresponding topological invariants can then be applied to the setting of Floquet metamaterials. In this paper these general results are considered in the case of a hexagonal structure. We provide two interesting examples of topologically non-trivial time-modulated hexagonal structures. [ABSTRACT FROM AUTHOR]

Multiscale high-contrast media can cause astonishing wave propagation phenomena through resonance effects. For instance, waves could be exponentially damped independent of the incident angle or waves could be re-focused as through a lense. In this review article, we discuss the numerical treatment of wave propagation through multiscale high-contrast media at the example of the Helmholtz equation. First, we briefly summarize the findings of analytical homogenization theory, which inspire the design of numerical methods and indicate interesting regimes for simulation. In the main part, we discuss two different classes of numerical multiscale methods and focus on how to treat especially high-contrast media. Some elements of a priori error analysis are discussed as well. Various numerical simulations showcase the applicability of the numerical methods to explore unusual wave phenomena, for instance exponential damping and lensing with flat interfaces. [ABSTRACT FROM AUTHOR]

The Navier equation is the governing equation of elastic waves, and computing its solution accurately and rapidly has a wide range of applications in geophysical exploration, materials science, etc. In this paper, we focus on the efficient and high-precision numerical algorithm for the time harmonic elastic wave scattering problems from cornered domains via the boundary integral equations in two dimensions. The approach is based on the combination of Nyström discretization, analytical singular integrals and kernel-splitting method, which results in a high-order solver for smooth boundaries. It is then combined with the recursively compressed inverse preconditioning (RCIP) method to solve elastic scattering problems from cornered domains. Numerical experiments demonstrate that the proposed approach achieves high accuracy, with stabilized errors close to machine precision in various geometric configurations. The algorithm is further applied to investigate the asymptotic behavior of density functions associated with boundary integral operators near corners, and the numerical results are highly consistent with the theoretical formulas. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the existence of solutions for a class of elliptic systems involving nonlinearities of the Keller-Osserman type and combined with the convection terms. Firstly, we establish a result involving sub and super-solution for a class of elliptic systems whose nonlinearity may depend of the gradient of the solution. This result permits the study of the existence of blow-up solution for a large class of systems. [ABSTRACT FROM AUTHOR]

In this work we prove, under symmetry and convexity assumptions on the domain Ω , the non- degeneracy at zero of the Hessian matrix of the Robin function for the spectral fractional Laplacian. This work extends to the fractional setting the results of M. Grossi concerning the classical Laplace operator. [ABSTRACT FROM AUTHOR]

Using variational methods, the existence and multiplicity of weak solutions for the following Neumann anisotropic problem [ABSTRACT FROM AUTHOR]

In this paper a second order elliptic equation with nonsmooth coefficients is considered in grand-Sobolev classes W q) 2 Ω on a bounded n-dimensional domain Ω ⊂ R n with a sufficiently smooth boundary ∂ Ω , generated by the norm of the grand-Lebesgue space L q) Ω . These spaces are non-separable and therefore the definition of a reasonable solution in them faces certain difficulties. For this purpose, a subspace N q) 2 Ω is distinguished in which infinitely differentiable and finite functions are dense. The strict inclusion W q 2 Ω ⊂ N q) 2 Ω holds, where W q 2 Ω is the classical Sobolev space. This raises specific questions dictated by the theory of spaces W q 2 Ω , for example, the characterization of the space of traces of functions from N q) 1 Ω cannot be characterized following the classical case. In this paper, the corresponding theorems concerning traces, extensions, and compactness of a family of functions from N q) k Ω are proved. These results are applied to obtain a Schauder-type estimate up to the boundary. Schauder-type estimates make it possible to establish the fredholmness of the Dirichlet problem for the considered equation in spaces N q) 2 Ω with data from grand-Lebesgue type spaces that are different from Lebesgue spaces. Therefore, the results of this work cannot be directly obtained from the results of the L p -theory. This work is a continuation of the research carried out by the authors in articles (Bilalov and Sadigova in Complex Var Elliptic Equ, 2020. https://doi.org/10.1080/17476933.2020.1807965; Bilalov and Sadigova in Sahand Commun Math Anal, 2021. https://doi.org/10.22130/scma.2021.521544.893. [ABSTRACT FROM AUTHOR]

In this paper, we provide an analytical study of the transmission eigenvalue problem with two conductivity parameters. We will assume that the underlying physical model is given by the scattering of a plane wave for an isotropic scatterer. In previous studies, this eigenvalue problem was analyzed with one conductive boundary parameter whereas we will consider the case of two parameters. We prove the existence and discreteness of the transmission eigenvalues as well as study the dependence on the physical parameters. We are able to prove monotonicity of the first transmission eigenvalue with respect to the parameters and consider the limiting procedure as the second boundary parameter vanishes. Lastly, we provide extensive numerical experiments to validate the theoretical work. [ABSTRACT FROM AUTHOR]

In this article, we mainly study the critical points of solutions to the Laplace equation with Dirichlet boundary conditions in an exterior domain in ℝ2. Based on the fine analysis about the structures of connected components of the super-level sets {x ∈ ℝ2 Ω: u(x) > t} and sub-level sets {x ∈ ℝ2 Ω: u(x) < t} for some t, we get the geometric distributions of interior critical point sets of solutions. Exactly, when Ω is a smooth bounded simply connected domain, u ∣ ∂ Ω = ψ (x) , lim ∣ x ∣ → ∞ u (x) = − ∞ and ψ(x) has K local maximal points on ∂Ω, we deduce that ∑ i = 1 l m i ≤ K , where m1, ..., ml; are the multiplicities of interior critical points x1, ..., xl; of solution u respectively. In addition, when ψ(x) has only K global maximal points and K equal local minima relative to ℝ2 Ω on ∂Ω, we have that ∑ i = 1 l m i = K . Moreover, when Ω is a domain consisting of l disjoint smooth bounded simply connected domains, we deduce that ∑ x i ∈ Ω m i + 1 2 ∑ x j ∈ ∂ Ω m j = l − 1 , and the critical points are contained in the convex hull of the l simply connected domains. [ABSTRACT FROM AUTHOR]

The numerical solution of the Helmholtz equations is challenging to compute when the wave numbers contained in the governing equation are large. In this paper, we present a parallel algorithm for this problem. A class of sixth-order hybrid compact finite-difference schemes for the Helmholtz equations is presented based on the Taylor expansion. To improve the efficiency of solving the large-wave-number problem, we implemented a parallel algorithm based on the Message Passing Interface environment to solve the discrete system. The validity and accuracy of the proposed method are verified by numerical examples. The method is also applicable to solving problems with oscillatory solutions, which are characterized by numerical instability as the wave number increases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps. [ABSTRACT FROM AUTHOR]

In this note, we give a simple proof of the pointwise BMO estimate for Poisson's equation. Then the Calderón–Zygmund estimate follows by the interpolation and duality. [ABSTRACT FROM AUTHOR]

In solving the coupled vapor and liquid unidirectional flows in micro heat pipes, we discovered numerically an integral identity. After asymptotic and polynomial expansions, the coupled flows yield two reciprocal systems of equations. In system A, a vapor velocity UA obeys the Poisson equation and drives, through an interfacial boundary condition, a liquid velocity WA that satisfies the Laplace equation. In reciprocal system B, a liquid velocity WB obeys the Poisson equation and drives, through another interfacial boundary condition, a vapor velocity UB that satisfies the Laplace equation. We found that the vapor volume flow rate of UB is numerically equal to the liquid volume flow rate of WA for seven different pipe shapes. Here, a general proof is presented for the integral identity, and some interesting implications of this identity are discussed.

In this paper, we find some superharmonic functions, which relate to the convexity estimates for Green's function and the first eigenfunction of Laplace operator with homogeneous Dirichlet boundary conditions in bounded convex domains of Rn. [ABSTRACT FROM AUTHOR]

We study the family of integral equations, called the Robin mean value equations (RMV), that are local averaged approximations to the Robin-Laplace boundary value problem (RL). When posed on C 1 , 1 -regular domains, we prove existence, uniqueness, equiboundedness and the comparison principle for solutions to (RMV). For the continuous right hand side of (RL), we show that solutions to (RMV) converge uniformly, in the limit of the vanishing radius of averaging, to the unique W2,p solution, which coincides with the unique viscosity solution of (RL). We further prove the lower bound on solutions to (RMV), which is consistent with the optimal lower bound for solutions to (RL). Our proofs employ martingale techniques, where (RMV) is interpreted as the dynamic programming principle along a suitable discrete stochastic process, interpolating between the reflecting and the stopped-at-exit Brownian walks. [ABSTRACT FROM AUTHOR]

We show that solutions to the Robin mean value equations (RMV), introduced in Lewicka and Peres (2022), converge uniformly in the limit of the vanishing radius of averaging, to the unique solution of the Robin-Laplace boundary value problem (RL), posed on any C 1 , 1 -regular domain and with any bounded Borel right hand side. When compared with the case of continuous right hand side, analyzed in Lewicka and Peres (2022), the present more general setting presents significant technical challenges. Along the way, we prove the asymptotic Hölder equicontinuity of solutions to (RMV): Lipschitz in the interior and C 0 , α up to the boundary, for any α ∈ (0,1). Our proofs employ martingale techniques, where (RMV) is interpreted as the dynamic programming principle for a discrete stochastic process, interpolating between the reflecting and the stopped-at-exit Brownian walks. [ABSTRACT FROM AUTHOR]

We consider the following Schrödinger equation with combined Hartree type and square-root nonlinearities - ▵ u = λ u + μ I α ∗ | u | p | u | p - 2 u + 1 - 1 1 + u 2 u , in R N , having prescribed mass ∫ R N | u | 2 = c , where N ≥ 2 and c > 0 is a given real number, λ appears as a Lagrange multiplier. Under the assumption of μ > 0 and N + α N < p < N + α + 2 N , we prove the existence of the normalized ground state by combining Concentration-compactness principle and estimate on the square-root nonlinearity. The main results extend and complement the earlier works. [ABSTRACT FROM AUTHOR]

In this paper, we use the fractional Legendre transform to treat the mixed boundary value problems on the unite sphere. We study the Laplace and Helmholtz operators on the unite sphere and coincide these operators with the fractional Sturm–Liouville problems. We apply the fractional Legendre transform to establish series approximations for the solutions of these problems. To obtain the unknown parts of solutions, we derive the Fredholm integral equations with the infinite series as separable kernels. We employ a technique for the integral equations to approximate the solutions in terms of the resolvent kernels. [ABSTRACT FROM AUTHOR]