On uniqueness and successive approximations in the Darboux problem for the equation $u_{xy} = f(x, y, u, u_x, u_y, ∫^x_0 ∫^y_0 g(x, y, s, t, u(s, t), u_s(s, t), u_t(s, t)) ds dt)$

B. Palczewski

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On the fundamental solution of the equation $∆^{p} u (X) + k (r) u (X) = 0$

J. Musiałek (1969)

Annales Polonici Mathematici

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L₁-uniqueness of degenerate elliptic operators

Derek W. Robinson, Adam Sikora (2011)

Studia Mathematica

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Let Ω be an open subset of $ℝ^{d}$ with 0 ∈ Ω. Furthermore, let $H_{Ω} = - \sum_{i, j = 1}^{d} \partial_{i} c_{i j} \partial_{j}$ be a second-order partial differential operator with domain $C_{c}^{\infty} (Ω)$ where the coefficients $c_{i j} \in W_{l o c}^{1, \infty} (Ω ̅)$ are real, $c_{i j} = c_{j i}$ and the coefficient matrix $C = (c_{i j})$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If $\int_{0}^{\infty} d s s^{d / 2} e^{- λ μ (s) ²} < \infty$ for some λ > 0 where $μ (s) = \int_{0}^{s} d t c {(t)}^{- 1 / 2}$ then we establish that $H_{Ω}$ is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup....