Recovering a variable exponent
We consider an inverse problem of recovering the non-linearity in the one dimensional variable exponent $p(x)$-Laplace equation from the Dirichlet-to-Neumann map. The variable exponent can be recovered up to the natural obstruction of rearrangements. The main technique is using the properties of a moment problem after reducing the inverse problem to determining a function from its $L^p$-norms.
Enclosure method for the p-Laplace equation
We study the enclosure method for the p-Calder\'on problem, which is a nonlinear generalization of the inverse conductivity problem due to Calder\'on that involves the p-Laplace equation. The method allows one to reconstruct the convex hull of an inclusion in the nonlinear model by using exponentially growing solutions introduced by Wolff. We justify this method for the penetrable obstacle case, where the inclusion is modelled as a jump in the conductivity. The result is based on a monotonicity inequality and the properties of the Wolff solutions.
Optimal recovery of a radiating source with multiple frequencies along one line
We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely determined by these measurements up to averaging over levelsets of the integrated attenuation. This leads to a generalized Laplace transform. We also discuss some numerical approaches and demonstrate the results with several examples.
Hamilton-Jacobi equations
Tämä Pro gradu-tutkielma käsittelee Hamiltonin ja Jacobin yhtälöitä, jotka kuvaavat mekaanisen järjestelmän kehitystä klassisen mekaniikan puitteissa. Hamiltonin ja Jacobin yhtälöitä käytetään myös säätöteoriassa sekä kvanttimekaniikassa. Hamiltonin mekaaniikan kehitti Sir William Rowan Hamilton valon käytöksen mallintamiseen ja Carl Gustav Jacob Jacobi kehitti sitä edelleen. Tutkielmassa annamme ehdot, joiden nojalla Hopfin ja Laxin kaava antaa ratkaisun Hamiltonin ja Jacobin yhtälöihin liittyvään alkuarvo-ongelmaan. Sen jälkeen määritämme sopivan heikon ratkaisun käsitteen ja näytämme heikkojen ratkaisujen olevan yksikäsitteisiä tietyillä ehdoilla. Lähestymme Hamiltonin ja Jacobin alkuarv…
Calder\'on's problem for p-Laplace type equations
We investigate a generalization of Calder\'on's problem of recovering the conductivity coefficient in a conductivity equation from boundary measurements. As a model equation we consider the p-conductivity equation with p strictly between one and infinity, which reduces to the standard conductivity equation when p equals two, and to the p-Laplace equation when the conductivity is constant. The thesis consists of results on the direct problem, boundary determination and detecting inclusions. We formulate the equation as a variational problem also when the conductivity may be zero or infinity in large sets. As a boundary determination result we recover the first order derivative of a smooth co…
Calderón problem for the p-Laplace equation : First order derivative of conductivity on the boundary
We recover the gradient of a scalar conductivity defined on a smooth bounded open set in Rd from the Dirichlet to Neumann map arising from the p-Laplace equation. For any boundary point we recover the gradient using Dirichlet data supported on an arbitrarily small neighbourhood of the boundary point. We use a Rellich-type identity in the proof. Our results are new when p 6 = 2. In the p = 2 case boundary determination plays a role in several methods for recovering the conductivity in the interior. peerReviewed
Superconductive and insulating inclusions for linear and non-linear conductivity equations
We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to prove partial results when the underlying equation is the quasilinear $p$-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation $\operatorname{div}(\sigma\lvert\nabla u\rvert^{p-2}\nabla u)=0$ where the measurable conductivity $\sigma\colon\Omega\to[0,\infty]$ is zero or infinity in large sets and $1<p<\infty$.
A posteriori-virhearvio Uzawan algoritmille Stokesin yhtälön ratkaisemiseksi
Tässä tutkielmassa esitän ensin lyhyesti Sobolevin avaruudet, totean joitakin epäyhtälöitä ja Stokesin yhtälön ajasta riippumattoman, englanniksi stationary, version. Jatkan todistamalla ratkaisun löytymisen Stokesin ongelmaan etsimällä Lagrangen funktion satulapisteen -- tämän todistuksen yksityiskohdat olen tarkastanut itse, vaikka todistus seuraakin hyvin tarkasti S. Repinin kirjasta \cite{repin2008} löytyvää tekstiä. Kappaleessa \ref{aposterioriarvio} kerron lyhyesti a~posteriori-tyylisen virhearvion luonteesta sekä totean erään Stokesin yhtälöä koskevan arvion. Seuraavassa kappaleessa esitän Uzawan algoritmin suoraan Stokesin yhtälöä varten. Viimeisessä kappaleessa esitän vihdoin kaksi…
Calderón's problem for p-laplace type equations
We investigate a generalization of Calderón’s problem of recovering the conductivity coefficient in a conductivity equation from boundary measurements. As a model equation we consider the p-conductivity equation div σ |∇u|p−2 ∇u = 0 with 1 < p < ∞, which reduces to the standard conductivity equation when p = 2. The thesis consists of results on the direct problem, boundary determination and detecting inclusions. We formulate the equation as a variational problem also when the conductivity σ may be zero or infinity in large sets. As a boundary determination result we recover the first order derivative of a smooth conductivity on the boundary. We use the enclosure method of Ikehata to recover the…
Monotonicity and enclosure methods for the p-Laplace equation
We show that the convex hull of a monotone perturbation of a homogeneous background conductivity in the $p$-conductivity equation is determined by knowledge of the nonlinear Dirichlet-Neumann operator. We give two independent proofs, one of which is based on the monotonicity method and the other on the enclosure method. Our results are constructive and require no jump or smoothness properties on the conductivity perturbation or its support.