Exact solutions of the boundary-value problems for the Helmholtz equation in a layer with polynomials in the right-hand sides of the equation and of the boundary conditions
In a multidimensional infinite layer bounded by two hyperplanes, the inhomogeneous Helmholtz equation with a polynomial right-hand side is considered. It is shown that the Dirichlet and Dirichlet-Neumann boundary-value problems with polynomials in the right-hand sides of the boundary conditions have a solution that is a quasipolynomial that contains, in addition to the power functions, also hyperbolic or trigonometric functions. This solution is unique in the class of slow growth functions if the parameter of the equation is not an eigenvalue. An algorithm for constructing this solution is given and examples are considered.
We prove new existence results for a Nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$ - Δu - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}) $$ with $k>0,$ $N \geq 3$, $p \in \left[\left.\frac{2(N+1)}{N-1},\frac{2N}{N-2}\right)\right.$ and $Q \in L^{\infty}(\mathbb{R}^{N})$. Due to the sign-changes of $Q$, our solutions have infinite Morse-Index in the corresponding dual variational formulation.
This paper presents local and global bifurcation results for radially symmetric solutions of the cubic Helmholtz system \begin{equation*} \begin{cases} -Δu - μu = \left( u^2 + b \: v^2 \right) u &\text{ on } \mathbb{R}^3, \\ -Δv - νv = \left( v^2 + b \: u^2 \right) v &\text{ on } \mathbb{R}^3. \end{cases} \end{equation*} It is shown that every point along any given branch of radial semitrivial solutions $(u_0, 0, b)$ or diagonal solutions $(u_b, u_b, b)$ (for $μ= ν$) is a bifurcation point. Our analysis is based on a detailed investigation of the oscillatory behavior of solutions at infinity t…
This paper considers a pair of coupled nonlinear Helmholtz equations \begin{align*} -Δu - μu = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*} -Δv - νv = a(x) \left( |v|^\frac{p}{2} + b(x) |u|^\frac{p}{2} \right)|v|^{\frac{p}{2} - 2}v \end{align*} on $\mathbb{R}^N$ where $\frac{2(N+1)}{N-1} < p < 2^\ast$. The existence of nontrivial strong solutions in $W^{2, p}(\mathbb{R}^N)$ is established using dual variational methods. The focus lies on necessary and sufficient conditions on the parameters deciding whether or not both components…
We derive simple-to-evaluate, closed-form solutions to the inhomogeneous Helmholtz equation, $Δu + k^2 u = χ_{B_{x_0,r}} $, the Schrödinger equation, $i\hbar \partial_t u + \frac{\hbar^2}{2m}Δu = 0$ with initial data ${u(x,0) = χ_{B_{x_0,r}} }$, and the Cauchy problem for the linear wave equation, ${\partial_t^2 u - c^2 Δu = 0 }$ with initial data $\left(u(x,0),\partial_t u(x,0)\right) = \left(χ_{B_{x_0,r}},χ_{B_{x_0,r}} \right). $ The function $χ_{B_{x_0,r}}$ is the characteristic function on the ball $B_{x_0,r} = \{x \in \mathbf{R}^3 : |x_0 - x| \leq r \} $. Furthermore, we use these solutio…
The focus of this paper is to show the absence of critical points for the solutions to the Helmholtz equation in a bounded domain $Ω\subset\mathbb{R}^{3}$, given by \[ \left\{ \begin{array}{l} -\rm{div}(a\,\nabla u_ω^{g})-ωqu_ω^{g}=0\quad\text{in $Ω$,}\\ u_ω^{g}=g\quad\text{on $\partialΩ$.} \end{array}\right. \] We prove that for an admissible $g$ there exists a finite set of frequencies $K$ in a given interval and an open cover $\overlineΩ=\cup_{ω\in K}Ω_ω$ such that $|\nabla u_ω^{g}(x)|>0$ for every $ω\in K$ and $x\inΩ_ω$. The set $K$ is explicitly constructed. If the spectrum of the above p…
Two boundary value problems for the Helmholtz equation in a semi-infinite strip are considered. The main feature of these problems is that, in addition to the function and its normal derivative on the boundary, the functionals of the boundary conditions possess tangential derivatives of the second and fourth orders. Also, the setting of the problems is complimented by certain edge conditions at the two vertices of the semi-strip. The problems model wave propagation in a semi-infinite waveguide with membrane and plate walls. A technique for the exact solution of these fluid-structure interactio…
We consider a quasilinear elliptic equation with right hand side measure, in which the lower order term has a behavior of jumping type. By means of techniques of degree theory, we prove the existence of one or two entropy solutions.
We introduce a new boundary Harnack principle in Lipschitz domains for equations with right hand side. Our approach, which uses comparisons and blow-ups, will adapt to more general domains as well as other types of operators. We prove the principle for divergence form elliptic equations with lower order terms including zero order terms. The inclusion of a zero order term appears to be new even in the absence of a right hand side.
We consider an one-phase free boundary problem for a degenerate fully non-linear elliptic operators with non-zero right hand side. We use the approach present in \cite{DeSilva} to prove that flat free boundaries and Lipschitz free boundaries are $C^{1, γ}$.
This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general "worst case" bound of exponential growth applies, and examples show that this growth can be realised through some sequence of wavenumbers, (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly conv…
We show that bounded solutions of the quasilinear elliptic equation $Δ_{p(x)} u=g+div(\textbf{F})$ are locally Hölder continuous provided that the functions $g$ and $\textbf{F}$ are in suitable Lebesgue spaces.
In this note, we consider complex Monge-Ampere equation posed on a compact Kähler manifold. We show how to get $L^p$($p<\infty$) and $L^{\infty}$ estimate for the gradient of the solution in terms of the continuity of the right hand side.
By developing new techniques we establish local existence and uniqueness theorems for an initial value problem involving a nonlinear equation in the sense of Riemann-Liouville fractional derivative in the case that the nonlinear function on the right hand side of the equation is not continuous on $[0,T]\times\mathbb{R}.$
We consider Monge-Ampère equations with right hand side $f$ that degenerate to $\infty$ near the boundary of a convex domain $Ω$, which are of the type $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;Ω,\quad\quad f\sim d^{-α}_{\partialΩ}\quad\mathrm{near}\;\partialΩ,$$ where $d_{\partialΩ}$ represents the distance to $\partial Ω$ and $-α$ is a negative power with $α\in(0,2)$. We study the boundary regularity of the solutions and establish a localization theorem for boundary sections.
The initial value problem for a coupled system is studied. The system consists of a differential inclusion and a differential equation and models the fluid flow of a viscoelastic fluid of Oldroyd type. The set-valued right-hand side of the differential inclusion satisfies certain measurability, continuity and growth conditions. The local existence (and global existence for small data) of a strong solution to the coupled system is shown using a generalisation of Kakutani's fixed-point theorem and applying results from the single-valued case.
We consider Monge-Ampére equations with the right hand side function close to a constant and from a function class that is larger than any Hölder class and smaller than the Dini-continuous class. We establish an upper bound for the modulus of continuity of the solution's second derivatives. This bound depends exponentially on a quantity similar to but larger than the Dini semi-norm. We establish explicit control on the shape of the sequence of shrinking sections, hence revealing the nature of such exponential dependence.
We establish sharp $C^{2s}$ interior regularity estimates for solutions of fully nonlinear nonlocal equations with bounded right hand side. More precisely, we show that if $I$ is a fully nonlinear nonlocal concave or convex elliptic operator and $f\in L^\infty(B_1)$ then \[ Iu=f\quad\textrm{ in }\quad B_1 \quad \Rightarrow\quad u\in C^{2s}(B_{1/2}). \] This result generalizes the linear counterpart proved by Ros-Oton and Serra and extends previous available results for fully nonlinear nonlocal operators. As an application, we get a basic regularity estimate for the nonlocal two membranes probl…
We establish global Hölder estimates for solutions to inhomogeneous linearized Monge-Ampère 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-Ampère measure being bounded away from zero and infinity. They are an affine invariant and degenerate version of global Hölder est…
Mixed variational approach to finding guaranteed estimates from solutions and right-hand sides of the second-order linear elliptic equations under incomplete data
Lipschitz regularity of solutions to two-phase $p$-Laplacian free boundary problems with right hand side
Dipartimento di Matematica dell’Università di Bologna（博洛尼亚大学数学系）；IMAS - CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires（布宜诺斯艾利斯大学理学院数学系、CONICET-IMAS）
In this article we obtain Holder estimates for solutions to second-order Hamilton-Jacobi equations with super-quadratic growth in the gradient and unbounded source term. The estimates are uniform with respect to the smallness of the diffusion and the smoothness of the Hamiltonian. Our work is in the spirit of a result by P. Cardaliaguet and L. Silvestre [5]. We utilize De Giorgi's method, which was introduced to this class of equations in [6].
We establish in this paper \emph{a priori} global $W^{2,δ}$ estimates for singular fully nonlinear elliptic equations with $L^n$ right hand side terms. The method is to slide paraboloids and barrier functions vertically to touch the solution of the equation, and then to estimate the measure of the contact set in terms of the measure of the vertex point set. To derive global estimates from $L^n$ data, the Hardy-Littlewood maximal functions, appropriate localizations and a new type of covering argument are adopted. These methods also provide us a more direct proof of the $W^{2,δ}$ estimates for …
We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from $L^2(\partial Ω)\rightarrow H^1(\partial Ω)$ (where $\partialΩ$ is the boundary of the obstacle). The new bounds are obtained using estimates on the restriction to the boundary of quasimodes of the Laplacian, building on recent work by the first author and collaborators. Our main motivation for considering these operators is that they appear in the standard second-kind boundary-integral formulations, posed in $L^2(\partial Ω)$, of the exterior Diri…
Regularity of flat free boundaries for a $p(x)$-Laplacian problem with right hand side
Dipartimento di Matematica dell’Università di Bologna（博洛尼亚大学数学系）；IMAS - CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires（布宜诺斯艾利斯大学精确与自然科学学院数学系、CONICET IMAS研究所）
This article is about the convex solution $u$ of the Monge--Ampère equation on an at least 2-dimensional open bounded convex domain with Dirichlet boundary data and nonnegative bounded right-hand side. For convex functions with zero boundary data, an Alexandrov maximum principle $|u(x)| \leq C \operatorname{dist}(x,\partialΩ)^α$ is equivalent to (uniform) Hölder continuity with the same constant and exponent. Convex $α$-Hölder continuous functions are $W^{1,p}$ for $p < 1/(1{-}α)$. We prove Hölder continuity with the exponent $α=2/n$ for $n \geq 3$ and any $α\in (0,1)$ for $n=2$, provided that…
