RSS 1.0Mathematics and Its Applications
Recent Submissions
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Smoothness in the 𝐿𝑝 Minkowski Problem for 𝑝<1We discuss the smoothness and strict convexity of the solution of the L p-Minkowski problem when p < 1 and the given measure has a positive density function.
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New competition phenomena in Dirichlet problemsWe study the multiplicity of nonnegative solutions to the problem, (Pλ) where Ω is a smooth bounded domain in RN, f:[0,∞)→R oscillates near the origin or at infinity, and p>0, λ∈R. While oscillatory right-hand sides usually produce infinitely many distinct solutions, an additional term involving up may alter the situation radically. Via a direct variational argument we fully describe this phenomenon, showing that the number of distinct non-trivial solutions to problem (Pλ) is strongly influenced by up and depends on λ whenever one of the following two cases holds: •p⩽1 and f oscillates near the origin; •p⩾1 and f oscillates at infinity (p may be critical or even supercritical). The coefficient a∈L∞(Ω) is allowed to change its sign, while its size is relevant only for the threshold value p=1 when the behaviour of f(s)/s plays a crucial role in both cases. Various - and L∞-norm estimates of solutions are also given.
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Elliptic-like regularization of semilinear evolution equationsConsider in a real Hilbert space the Cauchy problem (P0): u′(t)+Au(t)+Bu(t) = f (t), 0 ≤ t ≤ T ; u(0) = u_0, where −A is the generator of a C_0-semigroup of linear contractions and B is a smooth nonlinear operator. We associate with (P_0) the following problem: (Pε): −εu′′(t) + u′(t) + Au(t) + Bu(t) = f (t), 0 ≤ t ≤ T ; u(0) = u_0, u(T ) = u_1, where ε > 0 is a small parameter. Existence, uniqueness and higher regularity for both (P0) and (Pε) are investigated and an asymptotic expansion for the solution of problem (Pε) is established, showing the presence of a boundary layer near t = T .
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Eigenvalues of the Laplace operator with nonlinear boundary conditionsAn eigenvalue problem on a bounded domain for the Laplacian with a nonlinear Robin-like boundary condition is investigated. We prove the existence, isolation and simplicity of the first two eigenvalues.
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Eigenvalue problems in anisotropic Orlicz–Sobolev spacesWe establish sufficient conditions for the existence of solutions to a class of nonlinear eigenvalue problems involving nonhomogeneous differential operators in Orlicz–Sobolev spaces.
