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Register a new userInverse problems for real principal type operators
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We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determination methods for general operators, and prove global uniqueness results for determining coefficients in nonlinear real principal type equations. The article presents a unified approach for treating inverse boundary problems for transport and wave equations, and highlights the role of propagation of singularities in the solution of related inverse problems.
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We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.
Given two arbitrary sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$ of real numbers satisfying $$|lambda_1|>|mu_1|>|lambda_2|>|mu_2|>...>| lambda_j| >| mu_j| to 0 ,$$ we prove that there exists a unique sequence $c=(c_n)_{ninZ_+}$, real valued, such that the Hankel operators $Gamma_c$ and $Gamma_{tilde c}$ of symbols $c=(c_{n})_{nge 0}$ and $tilde c=(c_{n+1})_{nge 0}$ respectively, are selfadjoint compact operators on $ell^2(Z_+)$ and have the sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$ respectively as non zero eigenvalues. Moreover, we give an explicit formula for $c$ and we describe the kernel of $Gamma_c$ and of $Gamma_{tilde c}$ in terms of the sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$. More generally, given two arbitrary sequences $(rho_j)_{jge 1}$ and $(sigma_j)_{jge 1}$ of positive numbers satisfying $$rho_1>sigma_1>rho_2>sigma_2>...> rho_j> sigma_j to 0 ,$$ we describe the set of sequences $c=(c_n)_{ninZ_+}$ of complex numbers such that the Hankel operators $Gamma_c$ and $Gamma_{tilde c}$ are compact on $ell ^2(Z_+)$ and have sequences $(rho_j)_{jge 1}$ and $(sigma_j)_{jge 1}$ respectively as non zero singular values.
We establish Ambrosetti--Prodi type results for viscosity and classical solutions of nonlinear Dirichlet problems for the fractional Laplace and comparable operators. In the choice of nonlinearities we consider semi-linear and super-linear growth cases separately. We develop a new technique using a functional integration-based approach, which is more robust in the non-local context than a purely analytic treatment.
In this article we present three robust instability mechanisms for linear and nonlinear inverse problems. All of these are based on strong compression properties (in the sense of singular value or entropy number bounds) which we deduce through either strong global smoothing, only weak global smoothing or microlocal smoothing for the corresponding forward operators, respectively. As applications we for instance present new instability arguments for unique continuation, for the backward heat equation and for linear and nonlinear Calderon type problems in general geometries, possibly in the presence of rough coefficients. Our instability mechanisms could also be of interest in the context of control theory, providing estimates on the cost of (approximate) controllability in rather general settings.
We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = psi(x)$ where $f$ is a natural differential operator, with a restricted domain $F$, on a manifold $X$. By natural we mean operators that arise intrinsically from a given geometry on $X$. An important point is that the equation need not be convex and can be highly degenerate. Furthermore, the inhomogeneous term can take values at the boundary of the restricted domain $F$ of the operator $f$. A simple example is the real Monge-Amp`ere operator ${rm det}({rm Hess},u) = psi(x)$ on a riemannian manifold $X$, where ${rm Hess}$ is the riemannian Hessian, the restricted domain is $F = {{rm Hess} geq 0}$, and $psi$ is continuous with $psigeq0$. A main new tool is the idea of local jet-equivalence, which gives rise to local weak comparison, and then to comparison under a natural and necessary global assumption. The main theorem applies to pairs $(F,f)$, which are locally jet-equivalent to a given constant coefficient pair $({bf F}, {bf f})$. This covers a large family of geometric equations on manifolds: orthogonally invariant operators on a riemannian manifold, G-invariant operators on manifolds with G-structure, operators on almost complex manifolds, and operators, such as the Lagrangian Monge-Amp`ere operator, on symplectic manifolds. It also applies to all branches of these operators. Complete existence and uniqueness results are established with existence requiring the same boundary assumptions as in the homogeneous case [10]. We also have results where the inhomogeneous term $psi$ is a delta function.
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