Mathematics and Its Applications
Recent Submissions

Smoothness in the 𝐿𝑝 Minkowski Problem for 𝑝<1We discuss the smoothness and strict convexity of the solution of the L pMinkowski problem when p < 1 and the given measure has a positive density function.

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 righthand 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 nontrivial 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.

Ellipticlike 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_0semigroup 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 .

Eigenvalues of the Laplace operator with nonlinear boundary conditionsAn eigenvalue problem on a bounded domain for the Laplacian with a nonlinear Robinlike boundary condition is investigated. We prove the existence, isolation and simplicity of the ﬁrst two eigenvalues.

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.