Let $\Omega\subset\mathbb R^2$ be a bounded smooth domain. We prove that the weak Dirichlet-to-Neumann map for $-\Delta+V$ uniquely determines every complex-valued potential $V\in L^p(\Omega)$, $p>1$, provided that zero is not a Dirichlet eigenvalue. This extends the previously known range $p>4/3$ to all $p>1$. The proof uses Bukhgeim's quadratic-phase solutions and an average of Alessandrini's identity over the phase center. After separating the Neumann-series tails, we show that each remaining Born term tends to zero. The fixed-order estimates combine bounds for the absolute kernels with oscillatory cancellation and a duality argument for the product of the two Cauchy transforms.
We prove that the Dirichlet-to-Neumann map associated to the Schr\"odinger operator $-\Delta+q$ on a planar bounded Lipschitz domain uniquely determines an arbitrary complex-valued potential $q\in L^p$, $p>1$, whenever zero is not a Dirichlet eigenvalue. Our argument uses Nachman's d-bar equation only at sufficiently l...
Let $H_0$ be the standard discrete Laplacian on $\mathbb Z^d$, $d\geq4$, let $R_0(z)=(H_0-z)^{-1}$, and let $p'$ denote the H\"older conjugate of $p$, with $p'=\infty$ when $p=1$. We establish uniform diagonal resolvent estimates from $\ell^p(\mathbb Z^d)$ to $\ell^{p'}(\mathbb Z^d)$ by proving local Fourier-decay boun...
We study the existence and multiplicity of sign-changing solutions of the semilinear elliptic equation
$$ -\Delta _g u + u = f(u) \quad \text {on } \mathbb {S}^2, $$
where
$$(\mathbb {S}^2,g)$$
denotes the two-dimensional unit sphere endowed with a smooth Riemannian metric,
$$\Delta _g$$
is the La...
Manassés X. de Souza· Annali di Matematica Pura ed...· 0 citations
Let $n\ge2$, $1p-1$. We prove that every globally bounded fractional $p$-harmonic function is locally $C^{1,\alpha}$ for some $\alpha=\alpha(n,p,s)>0$. This settles the open problem of interior gradient H\"older regularity in the singular range throughout the natural first-order regime $sp>p-1$. The proof combines an a...
While unconditional uniqueness for the cubic nonlinear Schr\"odinger equation (NLS) on $\Bbb R^3$ is known in the scaling-subcritical Sobolev spaces $\dot H^s(\Bbb R^3)$ for $1/2<s<1$, the critical case $s=1/2$ has remained open. We resolve this problem by adapting Kato's bootstrap argument to a new choice of auxiliary...
We prove dispersive estimates for the three-dimensional defocusing energy-critical nonlinear Schr\"odinger equation associated with $\mathcal L_a=-\Delta+a|x|^{-2}$. For nonnegative potentials, we extend the known finite-$p$ theory to the endpoint $L^1\to L^\infty$. For negative potentials in the global well-posedness...
Tie-Song Jiang, Kexue Li· 0 citations
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