Teemu Lukkari

Unbounded Supersolutions of Nonlinear Equations with Nonstandard Growth

We show that every weak supersolution of a variable exponent-Laplace equation is lower semicontinuous and that the singular set of such a function is of zero capacity if the exponent is logarithmically Hölder continuous. As a technical tool we derive Harnack-type estimates for possibly unbounded supersolutions.

Vowel formants from the wave equation

This article describes modal analysis of acoustic waves in the human vocal tract while the subject is pronouncing [o]. The model used is the wave equation in three dimensions, together with physically relevant boundary conditions. The geometry is reconstructed from anatomical MRI data obtained by other researchers. The computations are carried out using the finite element method. The model is validated by comparing the computed modes with measured data.

A curious equation involving the ∞-Laplacian

Abstract We prove the uniqueness of viscosity solutions to a differential equation involving the infinity-Laplacian with a variable exponent. We also derive a version of Harnacks inequality for this minimax problem.

EQUIVALENCE OF VISCOSITY AND WEAK SOLUTIONS FOR THE p(x)-LAPLACIAN

Abstract We consider different notions of solutions to the p ( x ) -Laplace equation − div ( | D u ( x ) | p ( x ) − 2 D u ( x ) ) = 0 with 1 p ( x ) ∞ . We show by proving a comparison principle that viscosity supersolutions and p ( x ) -superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Rado type removability theorem.

Global regularity and stability of solutions to elliptic equations with nonstandard growth

We study the regularity properties of solutions to elliptic equations similar to the p(·)-Laplacian. Our main results are a global reverse Hölder inequality, Hölder continuity up to the boundary and stability of solutions with respect to continuous perturbations in the variable growth exponent. We assume that the complement of the domain is uniformly fat in a capacitary sense. As technical tools, we derive a capacitary Sobolev–Poincaré inequality, and a version of Hardys inequality.

Recording speech during magnetic resonance imaging

We discuss recording arrangements for speech during an MRI scan of the speakers vocal tract. The image and sound data thus obtained will be used for construction and validation of a numerical model for the vocal tract.

Perron's method for the porous medium equation

O. Perron introduced his celebrated method for the Dirichlet problem for harmonic functions in 1923. The method produces two solution candidates for given boundary values, an upper solution and a lower solution. A central issue is then to determine when the two solutions are actually the same function. The classical result in this direction is Wiener’s resolutivity theorem: the upper and lower solutions coincide for all continuous boundary values. We discuss the resolutivity theorem and the related notions for the porous medium equation ut −∆u = 0

ACOUSTIC WAVE GUIDES AS INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS

We prove the unique solvability, passivity/conservativity and some regularity results of two mathematical models for acoustic wave propagation in curved, variable diameter tubular structures of finite length. The first of the models is the generalised Websters model that includes dissipation and curvature of the 1D waveguide. The second model is the scattering passive, boundary controlled wave equation on 3D waveguides. The two models are treated in an unified fashion so that the results on the wave equation reduce to the corresponding results of approximating Websters model at the limit of vanishing waveguide intersection.

A comparison principle for the porous medium equation and its consequences

We prove a comparison principle for the porous medium equation in more general open sets in

A boundary estimate for singular parabolic diffusion equations

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