Continuity estimates for porous medium type equations with measure data - Bögelein, Verena; Duzaar, Frank; Gianazza, Ugo (Journal of Functional Analysis, 2014)
Keywords: Porous medium equation; Continuity; Riesz potential

Degenerate problems with irregular obstacles - Bögelein, Verena; Duzaar, Frank; Mingione, Giuseppe (Journal für die reine und angewandte Mathematik, 2011)
We establish the natural Calderón and Zygmund theory for solutions of elliptic and parabolic obstacle problems involving possibly degenerate operators in divergence form of p-Laplacian type, and proving that the (spatial) gradient of solutions is as integrable as that of the assigned obstacles. We also include an existence and regularity theorem where obstacles are not necessarily considered to be non-increasing in time.
UnivIS-Import:2015-03-09:Pub.2011.nat.dma.lma2.degene

Existence for evolutionary problems with linear growth by stability methods - Bögelein, Verena; Duzaar, Frank; Schätzler, Leah; Scheven, Christoph (Journal of Differential Equations, 2019)
We establish that solutions to the Cauchy–Dirichlet problem for functionals f:_ ×RN×n→[0, ∞)of linear growth can be obtained as limits of solutions to flows with p-growth in the limit p↓1. The result can be interpreted on the one hand as a stability result. On the other hand it provides an existence result for general flows with linear growth.∂tu −div _Dξf (x,Du)_ = 0<br
Keywords: Total variation flow; Flows with linear growth; Stability; Solutions obtained as limits of approximations

Existence of evolutionary variational solutions via the calculus of variations - Bögelein, Verena; Duzaar, Frank; Marcellini, Paolo (Journal of Differential Equations, 2014)
In this paper we introduce a purely variational approach to time dependent problems, yielding the existence of global parabolic minimizers, that is <a class="mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001144&_mathId=si1.gif&_user=111111111&_pii=S0022039614001144&_rdoc=1&_issn=00220396&md5=30ff8dbbb8f029d79a2a4fbfe1a9f265" title="View the MathML source"><img alt="View the MathML source" class="imgLazyJSB inlineImage" data-inlimgeid="1-s2.0-S0022039614001144-si1.gif" data-loaded="true" height="59" src="http://ars.els-cdn.com/content/image/1-s2.0-S0022039614001144-si1.gif" style="display: inline;" title="View the MathML source" width="409" /> <a class="mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001144&_mathId=si1.gif&_user=111111111&_pii=S0022039614001144&_rdoc=1&_issn=00220396&md5=30ff8dbbb8f029d79a2a4fbfe1a9f265" title="View the MathML source">whenever <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001144&_mathId=si2.gif&_user=111111111&_pii=S0022039614001144&_rdoc=1&_issn=00220396&md5=f318948ebe1bdd054b809b355ada5c43" title="Click to view the MathML source">T>0T>0 and φ∈<msubsup>C0∞</msubsup>(Ω×(0,T),RN). For the integrand <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001144&_mathId=si4.gif&_user=111111111&_pii=S0022039614001144&_rdoc=1&_issn=00220396&md5=a61b2670ec1d3b52e326606eb9a13f3a" title="Click to view the MathML source">f:Ω×RNn→[0,∞] we merely assume convexity with respect to the gradient variable and coercivity. These evolutionary variational solutions are obtained as limits of maps depending on space and time minimizing certain convex variational functionals. In the simplest situation, with some growth conditions on f, the method provides the existence of global weak solutions to Cauchy–Dirichlet problems of parabolic systems of the type <a class="mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001144&_mathId=si5.gif&_user=111111111&_pii=S0022039614001144&_rdoc=1&_issn=00220396&md5=2c9fdef4256f8d0ec83644f1edb173b0" title="View the MathML source"><img alt="View the MathML source" class="imgLazyJSB inlineImage" data-inlimgeid="1-s2.0-S0022039614001144-si5.gif" data-loaded="true" height="16" src="http://ars.els-cdn.com/content/image/1-s2.0-S0022039614001144-si5.gif" style="display: inline;" title="View the MathML source" width="287" />
Keywords: Existence; Parabolic systems; Parabolic minimizers; Evolutionary variational solutions; Evolutionary solutions via calculus of variations

Global solutions to the heat flow for m-harmonic maps and regularity - Bögelein, Verena; Duzaar, Frank; Scheven, Christoph (Indiana University Mathematics Journal, 2012)
UnivIS-Import:2015-03-09:Pub.2011.nat.dma.lma2.global

Gradient estimates via non standard potentials and continuity - Bögelein, Verena; Habermann, Jens (Annales Academiae Scientiarum Fennicae-Mathematica, 2010)
We consider elliptic problems with non standard growth conditions whose most prominent model example is the p(x)-Laplacean equation with a measure data right-hand side μ. We prove pointwise gradient estimates in terms of a non standard version of the non-linear Wolff potential of the right-hand side measure, and moreover a characterization for C1-regularity of the solution, also in terms of the Wolff potential. The C1-regularity criterion is also related to the density of μ and the decay rate of its L"-norm on small balls. Moreover, from the pointwise gradient estimates the Calderón and Zygmund theory and several types of local estimates follow as a consequence.
UnivIS-Import:2015-03-09:Pub.2010.nat.dma.lma2.gradie_4
Keywords: Measure data problems; Non standard wolff potential; Partial differential equations with non standard growth; Pointwise gradient estimates

Higher integrability for doubly nonlinear parabolic systems - Bögelein, Verena; Duzaar, Frank; Kinnunen, Juha; Scheven, Christoph (Journal De Mathematiques Pures Et Appliquees, 2020)
Abstract: This paper proves a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems. The new feature of the argument is that the intrinsic geometry involves the solution as well as its spatial gradient. The main result holds true for a range of parameters suggested by other nonlinear parabolic systems.<br

Higher integrability for parabolic systems with non-standard growth and degenerate diffusions - Bögelein, Verena; Duzaar, Frank (Publicacions Matematiques, 2011)
The aim of this paper is to establish a Meyer's type higher integrability result for weak solutions of possibly degenerate parabolic systems of the type ∂tu - div a(x; t;Du) = div - |F|p(x;t)-2F F The vector-éld a is assumed to fulfill a non-standard p(x; t)- growth condition. In particular it is shown that there exists ε > 0 depending only on the structural data such that there holds: |Du|p(·) (1+ε) ∈ Lloc 1; together with a local estimate for the p(·)(1 + ε)-energy.
UnivIS-Import:2015-03-09:Pub.2011.nat.dma.lma2.higher
Keywords: Degenerate parabolic systems; Higher integrability; Non-standard growth condition; Parabolic p-Laplacean

Holder estimates for parabolic p(x, t)-Laplacian systems - Bögelein, Verena; Duzaar, Frank (Mathematische Annalen, 2012)

In memoriam Thomas Friedrich (1949–2018). - Agricola, Ilka; Bögelein, Verena; Duzaar, Frank (Annals of Global Analysis and Geometry, 2019)
Thomas Friedrich, who founded the journal ‘Annals of Global Analysis and Geometry’ together with Rolf Sulanke in 1983 and acted as editor in chief for more than 3 decades, died in Marburg (Germany) on February 27, 2018, at the age of sixty-eight of COPD and lung cancer. Besides sketching Thomas’s biography and scientific work, it is our goal in this obituary to tell the founding story of ‘his’ journal. This will reflect his life and character, and will provide a remarkable insight into twentieth century science policies as well as the changes that scientific publishing has undergone in the past decade

Parabolic equations with (p,q)-growth - Bögelein, Verena; Duzaar, Frank; Marcellini, Paolo (Journal De Mathematiques Pures Et Appliquees, 2013)
UnivIS-Import:2015-03-09:Pub.2011.nat.dma.lma2.parabo_9

Parabolic Systems with p, q-Growth: A Variational Approach - Bögelein, Verena; Duzaar, Frank; Marcellini, Paolo (Archive for Rational Mechanics and Analysis, 2013)

Partial Hölder continuity for discontinuous elliptic problems with VMO-coefficients - Bögelein, Verena; Duzaar, Frank; Habermann, Jens; Scheven, Christoph (Proceedings of the London Mathematical Society, 2011)
We establish partial Hölder continuity for vector-valued solutions u: Ω → N to elliptic systems of the type as well as for minimizers u: Ω → N of quasi-convex functionals where the structure function a, respectively, the integrand f is possibly discontinuous with respect to x. More precisely, we merely impose a uniform VMO-condition with respect to the x-dependence and continuity with respect to the u-dependence and prove Hölder continuity of the solutions, respectively, the minimizers outside of a negligible set. 2011 London Mathematical Society2011 © 2011 London Mathematical Society.
UnivIS-Import:2015-03-09:Pub.2011.nat.dma.lma2.partia

POROUS MEDIUM TYPE EQUATIONS WITH MEASURE DATA AND POTENTIAL ESTIMATES - Bögelein, Verena; Duzaar, Frank; Gianazza, Ugo (SIAM Journal on Mathematical Analysis, 2013)
We consider nonhomogeneous, degenerate (m > 1) porous medium type equations with a nonnegative Radon measure mu having finite total mass mu(E-T) on the right-hand side, and we derive linear pointwise estimates for solutions via Riesz potentials. Then, we deal with a Cauchy Dirichlet problem for the same equation, with homogeneous boundary conditions on the parabolic boundary of the domain E-T, and we establish the existence of a solution in the sense of distributions.
Keywords: porous medium equation; measure data; existence; potential estimates

Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients - Bögelein, Verena; Duzaar, Frank; Habermann, Jens; Scheven, Christoph (Advances in Calculus of Variations, 2012)
We establish partial regularity for solutions (u, π): ℝ n ×ℝ n ⊇ Ω × Σ → ℝ n × ℝ to systems modeling electro-rheological fluids in the stationary case. As a model case our result covers the low order regularity of systems of the type (equation) where εu denotes the symmetric part of the gradient Du, π denotes the pressure, the not necessarily continuous coefficient μ is a bounded non-negative VMO-function and the variable exponent function p: Ω → (3n/n+2, ∞) fulfills the logarithmic continuity assumption, i.e., we assume that for the modulus of continuity ω p of the exponent function p there holds (equation) To be more precise, we prove Hölder continuity of the solution u outside of a negligible set. Moreover, we show that εu and the pressure belong to certain Morrey spaces on the regular set of u, i.e., the set where u is Hölder continuous. Note that under such weak assumptions partial Hölder continuity for the gradient cannot be expected. Our result is even new if the coefficient μ is continuous. © de Gruyter 2012.
UnivIS-Import:2015-03-09:Pub.2012.nat.dma.lma2.statio
Keywords: Electro-rheological fluids; Non-standard growth condition; Partial Hölder-and Morrey-type regularity; VMO-coefficients

The boundary regularity of non-linear parabolic systems I - Bögelein, Verena; Duzaar, Frank; Mingione, Giuseppe (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2010)
This is the first part of a work aimed at establishing that for solutions to Cauchy-Dirichlet problems involving general non-linear systems of parabolic type, almost every parabolic boundary point is a Hölder continuity point for the spatial gradient of solutions. Here we develop the basic necessary and sufficient condition for establishing the regular nature of a boundary point.
UnivIS-Import:2015-03-09:Pub.2010.nat.dma.lma2.thebou

The boundary regularity of non-linear parabolic systems II - Bögelein, Verena; Duzaar, Frank; Mingione, Giuseppe (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2010)
This is the second part of a work aimed at establishing that for solutions to Cauchy–Dirichlet problems involving general non-linear systems of parabolic type, almost every parabolic boundary point is a Hölder continuity point for the spatial gradient of solutions. Here we establish higher fractional differentiability of solutions up to the boundary. Based on the necessary and sufficient condition for regular boundary points from the first part of Bögelein et al. (in this issue) [7] we achieve dimension estimates for the boundary singular set and eventually the almost everywhere regularity of solutions at the boundary.<br

The regularity of general parabolic systems with degenerate diffusions - Bögelein, Verena; Duzaar, Frank; Mingione, Giuseppe (Memoirs of the American Mathematical Society, 2013)
UnivIS-Import:2015-03-09:Pub.2011.nat.dma.lma2.thereg

Weak solutions to the heat flow for surfaces of prescribed mean curvature - Bögelein, Verena; Duzaar, Frank; Scheven, Christoph (Transactions of the American Mathematical Society, 2013)
UnivIS-Import:2015-03-09:Pub.2012.nat.dma.lma2.weakso