We consider the optimization problem of minimizing R Ω G(|∇u|) + λχ {u>0} dx in the class of functions W 1,G (Ω) with u − ϕ0 ∈ W 1,G 0 (Ω), for a given ϕ0 ≥ 0 and bounded. W 1,G (Ω) is the class of weakly differentiable functions with R Ω G(|∇u|) dx < ∞. The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω ∩ ∂{u > 0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C 1,α regularity of their free boundaries near "flat" free boundary points.This kind of optimization problem has been widely studied for different functions G. In fact, the first paper in which this problem was studied is [3]. The authors considered the case G(t) = t 2 . They proved that minimizers are weak solutions to the free boundary problem (1.2) ∆u = 0 in {u > 0} u = 0, |∇u| = λ on ∂{u > 0} and proved the Lipschitz regularity of the solutions and the C 1,α regularity of the free boundaries. This free boundary problem appears in several applications. A very important one is that of fluid flow. In that context, the free boundary condition is known as Bernoulli's condition.The results of [3] have been generalized to several cases. For instance, in [5] the authors consider problem (1.1) for a convex function G such that ct < G ′ (t) < Ct for some positive constants c and C. Recently, in the article [7] the authors considered the case G(t) = t p with 1 < p < ∞. In these two papers only minimizers are studied. Minimizers satisfy very good properties like nondegeneracy at the free boundary and uniform positive density of the set {u = 0} at free boundary points. On the other hand, the free boundary problem (1.2) and its

The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order of such convergence. Finally, we perform some numerical experiments and compare our results with previous work by other authors.2010 Mathematics Subject Classification. 46E35, 35P15, 49R05 65N25, 65N30.

We prove the simplicity and isolation of the first eigenvalue for the problem ∆ p u = |u| p−2 u in a bounded smooth domain Ω ⊂ R N , with a nonlinear boundary condition given by |∇u| p−2 ∂u/∂ν = λ|u| p−2 u on the boundary of the domain.

Summary In this work, we present a new algorithm for wide‐field fluorescent micrsocopy deconvolution from a single acquisition without a sparsity prior, which allows the retrieval of the target function with superresolution, with a simple approach that the measured data are fit by the convolution of a superposition of virtual point sources (SUPPOSe) of equal intensity with the point spread function. The cloud of virtual point sources approximates the actual distribution of sources that can be discrete or continuous. In this manner, only the positions of the sources need to be determined. An upper bound for the uncertainty in the position of the sources was derived, which provides a criteria to distinguish real facts from eventual artefacts and distortions. Two very different experimental situations were used for the test (an artificially synthesized image and fluorescent microscopy images), showing excellent reconstructions and agreement with the predicted uncertainties, achieving up to a fivefold improvement in the resolution for the microscope. The method also provides the optimum number of sources to be used for the fit. Lay Description A new method is presented that allows the reconstruction of an image with superresolution from a single frame taken with a standard fluorescent microscope. An improvement in the resolution of a factor between 3 and 5 is achieved depending on the noise of the measurement and how precisely the instrument response function (point spread function) is measured. The complete mathematical description is presented showing how to estimate the quality of the reconstruction. The method is based in the approximation of the actual intensity distribution of the object being measured by a superposition of point sources of equal intensity. The problem is converted from determining the intensity of each point to determining the position of the virtual sources. The best fit is found using a genetic algorithm. To validate the method several results of different nature are presented including an artificially generated image, fluorescent beads and labelled mitochondria. The artificial image provides a prior knowledge of the actual system for comparison and validation. The beads were imaged with our highest numerical aperture objective to show method capabilities and also acquired with a low numerical aperture objective to compare the reconstructed image with that acquired with a high numerical aperture objective. This same strategy was followed with the biological sample to show the method working in real practical situations.