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We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that ∑j=1n Φ(λjA/G) is maximal for a disk whenever Φ is concave increasing, n≥1, the domain has area A, and λj is the j-th Dirichlet eigenvalue of the magnetic Laplacian (i∇+β/2A(-x2,x1))2. Here the flux β is constant, and the scale invariant factor G penalizes deviations from roundness, meaning G≥1 for all domains and G=1 for disks. [ABSTRACT FROM AUTHOR]

Every bounded linear operator that maps ${H}^{1} $ to ${L}^{1} $ and ${L}^{2} $ to ${L}^{2} $ is bounded from ${L}^{p} $ to ${L}^{p} $ for each $p\in (1, 2)$, by a famous interpolation result of Fefferman and Stein. We prove ${L}^{p} $-norm bounds that grow like $O(1/ (p- 1))$ as $p\downarrow 1$. This growth rate is optimal, and improves significantly on the previously known exponential bound $O({2}^{1/ (p- 1)} )$. For $p\in (2, \infty )$, we prove explicit ${L}^{p} $ estimates on each bounded linear operator mapping ${L}^{\infty } $ to bounded mean oscillation ($\mathit{BMO}$) and ${L}^{2} $ to ${L}^{2} $. This $\mathit{BMO}$ interpolation result implies the ${H}^{1} $ result above, by duality. In addition, we obtain stronger results by working with dyadic ${H}^{1} $ and dyadic $\mathit{BMO}$. The proofs proceed by complex interpolation, after we develop an optimal dyadic ‘good lambda’ inequality for the dyadic $\sharp $-maximal operator. [ABSTRACT FROM PUBLISHER]

The sum of the first n >= 1 eigenvalues of the Laplacian is shown to be maximal among simplexes for the regular simplex (the regular tetrahedron, in three dimensions), maximal among parallelepipeds for the hypercube, and maximal among ellipsoids for the ball, provided the volume and moment of inertia of an auxiliary body are suitably normalized. This result holds for Dirichlet, Robin, and Neumann eigenvalues. Additionally, the cubical torus is shown to be maximal among flat tori. The proof involves euclidean tight frames generated by orbits of the rotation group of the extremal domain. Among general convex domains, the ball is conjectured to maximize sums of Neumann eigenvalues, provided the volume and moment of inertia of the polar dual are suitably normalized. [ABSTRACT FROM AUTHOR]

We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proven for the harmonic and arithmetic means of the first two nonzero eigenvalues. [ABSTRACT FROM AUTHOR]

We investigate the problem of minimizing the moment of inertia among convex surfaces in R3 having a specified surface area. First, we prove that a minimizing surface exists, and derive a necessary condition holding at points of positive curvature. Then we show that an equilateral triangular prism is the optimal triangular prism, that the cube is the optimal rectangular prism, and that the sphere is (locally) optimal among ellipsoids. Many examples of convex surfaces are examined, among which the lowest moment of inertia is achieved by a truncated tetrahedron. The problem of finding the global minimizing surface remains open. The analogous problem in two dimensions has been solved by Sachs and later by Hall, who showed that the equilateral triangle minimizes the moment of inertia, among all convex curves with given length. [ABSTRACT FROM AUTHOR]

We study the phase space of the evolution equationht=−(f(h)hxxx)x−(g(h)hx)x by means of a dissipated energy. Here h(x,t)⩾0, and at h=0 the coefficient functions f>0 and g can either degenerate to 0, or blow up to ∞, or tend to a nonzero constant. We first show that all positive periodic steady-states are saddles in the energy landscape, with respect to zero-mean perturbations, if (g/ f)″⩾0 or if the perturbations are allowed to have period longer than that of the steady-state. For power-law coefficients (f (y)= yn and g(y)=Bym for some B>0) we analytically determine the relative energy levels of distinct steady-states. For example, with m−n∈[1,2) and for suitable choices of the period and mean value, we find three fundamentally different steady-states. The first is a constant steady-state that is stable and is a local minimum of the energy. The second is a positive periodic steady-state that is linearly unstable and has higher energy than the constant steady-state; it is a saddle point for the energy. The third is a periodic collection of ‘droplet’ (compactly supported) steady-states having lower energy than either the positive steady-state or the constant one. Since the energy must decrease along every orbit, these results significantly constrain the dynamics of the evolution equation. [Copyright &y& Elsevier]