Continuity up to the boundary for obstacle problems to porous medium type equations

We show that signed weak solutions to obstacle problems for porous medium type equations with Cauchy-Dirichlet boundary data are continuous up to the parabolic boundary, provided that the obstacle and boundary data are continuous. This result seems to be new even for signed solutions to the (obstacle free) Cauchy-Dirichlet problem to the singular porous medium equation, which is retrieved as a special case

Supercaloric functions for the porous medium equation in the fast diffusion case

We study a generalized class of supersolutions, so-called supercaloric functions to the porous medium equation in the fast diffusion case. Supercaloric functions are defined as lower semicontinuous functions obeying a parabolic comparison principle. We prove that bounded supercaloric functions are weak supersolutions. In the supercritical range, we show that unbounded supercaloric functions can be divided into two mutually exclusive classes dictated by the Barenblatt solution and the infinite point-source solution, and give several characterizations for these classes. Furthermore, we study the pointwise behavior of supercaloric functions and obtain connections between supercaloric functions and weak supersolutions.Comment: Corrected typographical errors and made minor notational adjustment

Existence of solutions for nonlinear parabolic problems via direct methods in the calculus of variations

Työssä todistetaan ratkaisun olemassaolo ja yksikäsitteisyys paraboliselle p-Laplacen yhtälölle ajasta riippumattomilla Cauchy-Dirichlet'n reuna-arvoilla. Tulos perustuu variaatiolaskennan suoriin menetelmiin, jolloin todistus nojaa oleellisesti vain yhtälön ratkaisun variaatiomääritelmään sekä energiaestimaatteihin ja approksimaatioargumentteihin, kuten aikasilotukseen. Kyseinen todistustekniikka on sovellettavissa myös yleisemmille parabolisille systeemeille.We prove existence and uniqueness of solution for parabolic p-Laplace equation with time independent Cauchy-Dirichlet boundary datum. The result is based on direct methods in the calculus on variations. In this case, the existence is achieved with variational definition of solution and by using energy estimates and approximation arguments, such as time mollification. Due to the methods used in the proof, the result can be applied to more general parabolic systems as well

Gradient higher integrability for degenerate parabolic double-phase systems

We prove local higher integrability of the gradient of a weak solution to a degenerate parabolic double-phase system. This result comes with a reverse H\"older type estimate for the gradient. The proof is based on estimates in the intrinsic geometry and stopping time arguments

Supercaloric functions for the parabolic pp-Laplace equation in the fast diffusion case

We study a generalized class of supersolutions, so-called $p$-supercaloric functions, to the parabolic $p$-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for $p\geq 2$, but little is known in the fast diffusion case $1<p<2$. Every bounded $p$-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic $p$-Laplace equation for the entire range $1<p<\infty$. Our main result shows that unbounded $p$-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case $\frac{2n}{n+1}<p<2$. The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case $1<p\leq \frac{2n}{n+1}$ and the theory is not yet well understood

ECLAIRE third periodic report

The ÉCLAIRE project (Effects of Climate Change on Air Pollution Impacts and Response Strategies for European Ecosystems) is a four year (2011-2015) project funded by the EU's Seventh Framework Programme for Research and Technological Development (FP7)

ECLAIRE: Effects of Climate Change on Air Pollution Impacts and Response Strategies for European Ecosystems. Project final report

The central goal of ECLAIRE is to assess how climate change will alter the extent to which air pollutants threaten terrestrial ecosystems. Particular attention has been given to nitrogen compounds, especially nitrogen oxides (NOx) and ammonia (NH3), as well as Biogenic Volatile Organic Compounds (BVOCs) in relation to tropospheric ozone (O3) formation, including their interactions with aerosol components. ECLAIRE has combined a broad program of field and laboratory experimentation and modelling of pollution fluxes and ecosystem impacts, advancing both mechanistic understanding and providing support to European policy makers. The central finding of ECLAIRE is that future climate change is expected to worsen the threat of air pollutants on Europe’s ecosystems. Firstly, climate warming is expected to increase the emissions of many trace gases, such as agricultural NH3, the soil component of NOx emissions and key BVOCs. Experimental data and numerical models show how these effects will tend to increase atmospheric N deposition in future. By contrast, the net effect on tropospheric O3 is less clear. This is because parallel increases in atmospheric CO2 concentrations will offset the temperature-driven increase for some BVOCs, such as isoprene. By contrast, there is currently insufficient evidence to be confident that CO2 will offset anticipated climate increases in monoterpene emissions. Secondly, climate warming is found to be likely to increase the vulnerability of ecosystems towards air pollutant exposure or atmospheric deposition. Such effects may occur as a consequence of combined perturbation, as well as through specific interactions, such as between drought, O3, N and aerosol exposure. These combined effects of climate change are expected to offset part of the benefit of current emissions control policies. Unless decisive mitigation actions are taken, it is anticipated that ongoing climate warming will increase agricultural and other biogenic emissions, posing a challenge for national emissions ceilings and air quality objectives related to nitrogen and ozone pollution. The O3 effects will be further worsened if progress is not made to curb increases in methane (CH4) emissions in the northern hemisphere. Other key findings of ECLAIRE are that: 1) N deposition and O3 have adverse synergistic effects. Exposure to ambient O3 concentrations was shown to reduce the Nitrogen Use Efficiency of plants, both decreasing agricultural production and posing an increased risk of other forms of nitrogen pollution, such as nitrate leaching (NO3-) and the greenhouse gas nitrous oxide (N2O); 2) within-canopy dynamics for volatile aerosol can increase dry deposition and shorten atmospheric lifetimes; 3) ambient aerosol levels reduce the ability of plants to conserve water under drought conditions; 4) low-resolution mapping studies tend to underestimate the extent of local critical loads exceedance; 5) new dose-response functions can be used to improve the assessment of costs, including estimation of the value of damage due to air pollution effects on ecosystems, 6) scenarios can be constructed that combine technical mitigation measures with dietary change options (reducing livestock products in food down to recommended levels for health criteria), with the balance between the two strategies being a matter for future societal discussion. ECLAIRE has supported the revision process for the National Emissions Ceilings Directive and will continue to deliver scientific underpinning into the future for the UNECE Convention on Long-range Transboundary Air Pollution

Säännöllisyysteoriaa epälineaarisille parabolisille osittaisdifferentiaaliyhtälöille: gradienttiestimaatit, stabiilisuus ja esteongelma

This thesis concerns different aspects of regularity theory for weak solutions of nonlinear parabolic partial differential equations. We focus on such equations with porous medium type and p-growth structure. In particular, we consider solutions which are defined either via a weak formulation of the equation with test functions under the integral sign, or as functions that obey a parabolic comparison principle. Our research concerns the regularity of both the solution and its gradient. For the gradient of a weak solution of porous medium type systems we prove a higher integrability result up to the boundary of the domain. We derive reverse Hölder inequalities in intrinsic cylinders near the boundary, for which we prove a Vitali type covering property that is applied to obtain the higher integrability result. We also show that under suitable assumptions, weak solutions as well as their gradients are stable with respect to small fluctuations of the parameter characterizing the equation. In particular, we prove that solutions to the approximating problems converge to the corresponding solution of the limit problem in the natural parabolic Sobolev space. For the parabolic p-Laplace equation we study supersolutions, which are defined via a parabolic comparison principle. We show that in the fast diffusion case these functions can be divided into two mutually exclusive classes, for which we give several characterizations. An important tool in regularity theory is the obstacle problem, which is also interesting in its own right. In the case of signed obstacles we study Hölder continuity for solutions to the porous medium type equations defined via a variational inequality. We use a De Giorgi type iteration argument to show that solutions to obstacle problems are locally Hölder continuous, provided that the obstacle is Hölder continuous.Väitöskirjassa tutkitaan säännöllisyysteoriaa eri näkökulmista epälineaaristen parabolisten osittaisdifferentiaaliyhtälöiden heikoille ratkaisuille. Tutkimus keskittyy yhtälöihin, joilla on huokoisen aineen tyyppiset tai p-kasvuehdot. Työssä tarkastellaan erityisesti ratkaisuja, jotka on joko määritelty yhtälön heikon muodon avulla testifunktioita vastaan integroiden, tai jotka kytke-tään yhtälöön käyttäen parabolista vertailuperiaatetta. Tutkimus kohdistuu sekä itse ratkaisun että ratkaisun gradientin säännöllisyyteen liittyviin kysymyksiin. Huokoisen aineen systeemien heikon ratkaisun gradientille todistetaan korkeampi integroituvuus alueen reunalle asti. Gradientille johdetaan käänteisen Hölderin epäyhtälöt lähellä alueen reunaa sellaisissa sylintereissä, joiden geometria on yhtälön rakenteelle ominainen. Kyseisille sylintereille osoitetaan Vitali-tyyppinen peitelause, jota sovelletaan korkeamman integroituvuuden todistuksessa. Väitöskirjassa näytetään myös, että asianmukaisilla oletuksilla sekä heikot ratkaisut että ratkaisujen gradientit ovat stabiileja ongelmaa karakterisoivan parametrin heilahtelujen suhteen. Työssä todistetaan, että approksi-moivien ongelmien ratkaisut suppenevat rajaongelman ratkaisuun luonnollisessa parabolisessa Sobolevin avaruudessa. Parabolisen p-Laplacen yhtälölle tutkitaan superratkaisuja, jotka määri-tellään parabolisen vertailuperiaatteen avulla. Työssä osoitetaan, että nopean diffuusion tapauk-sessa kyseiset superratkaisut voidaan jakaa kahteen erilliseen luokkaan, joille esitetään useita eri karakterisointeja. Esteongelma on tärkeä työkalu säännöllisyysteoriassa mutta myös kiinnostava tutkimuskohde itsessään. Variaatioepäyhtälön avulla määritellyn huokoisen aineen yhtälön este-ongelman ratkaisun Hölder-jatkuvuutta tutkitaan merkkiä vaihtavan esteen tapauksessa. De Giorgi -tyyppistä iteraatioargumenttia soveltaen työssä todistetaan ratkaisun lokaali Hölder-jatkuvuus esteen ollessa Hölder-jatkuva