Arttu Karppinen
We prove continuity up to the boundary of the minimizer of an obstacle problem and higher integrability of its gradient under generalized Orlicz growth. The result recovers similar results obtained in the special cases of polynomial growth, variable exponent growth and produces new results for Orlicz and double phase growth.
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We prove local $C^{0,α}$- and $C^{1,α}$-regularity for the local solution to an obstacle problem with non-standard growth. These results cover as special cases standard, variable exponent, double phase and Orlicz growth.
Lipschitz regularity of almost-minimizers in one-phase problems with generalized Orlicz growth
研究了单相问题中几乎极小值解的Lipschitz正则性，针对具有广义Orlicz增长的Alt-Caffarelli型泛函建立了最优局部正则性结果。
We study the following boundary value problem (P)\ \ \ \ \ {-\mathrm{div}(a(|\nabla u|)\nabla u)=f(x,u),\ & in $Ω$, u=0, & on $\partialΩ$} with nonhomogeneous principal part. By assuming the nonlinearity $f(x, t)$ being subcritical growth, some abstract results of problem (P) are obtained: (1) Regularity; (2) Orlicz-Sobolev versus Hölder local minimizer; (3) Strong comparison principle. Applying these abstract results and critical point theory, we prove the existence of multiple solutions of problem (P) in an Orlicz-Sobolev space.
We study minimizers of non-autonomous functionals \begin{align*} \inf_u \int_Ωφ(x,|\nabla u|) \, dx \end{align*} when $φ$ has generalized Orlicz growth. We consider the case where the upper growth rate of $φ$ is unbounded and prove the Harnack inequality for minimizers. Our technique is based on "truncating" the function $φ$ to approximate the minimizer and Harnack estimates with uniform constants via a Bloch estimate for the approximating minimizers.
Orlicz-Sobolev versus Hölder local minimizer for nonlinear Robin problems
本文针对 Orlicz $g$-拉普拉斯算子驱动的 Robin 问题，利用 Moser 迭代变体证明弱解有界，并结合 Lieberman 定理，建立了该问题能量泛函的 $C^1(\bar{\u03a9})$-局部极小值与 $W^{1,G}(\u03a9)$-局部极小值的等价关系。
Local boundedness of minimizers under unbalanced Orlicz growth conditions
本文研究积分泛函在非平衡 Orlicz 增长条件下的局部正则性，通过平衡被积函数的上下界证明极小化子与拟极小化子局部有界，并推广了 $p,q$-增长的经典尖锐结果。
Regularity to Thin Obstacle Problem in Orlicz spaces
本文研究了Orlicz空间中薄障碍问题的极小解正则性，证明了梯度的Lipschitz连续性和Hölder连续性，并揭示了极小解零点集的结构特征。
A note on the weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
本文在局部 L^s 可积条件下证明 De Giorgi 类函数的弱 Harnack 不等式，将结果推广至非对数条件下的变指数与双相位广义 Orlicz 增长泛函。
We show local Hölder continuity of quasiminimizers of functionals with non-standard (Musielak--Orlicz) growth. Compared with previous results, we cover more general minimizing functionals and need fewer assumptions. We prove Harnack's inequality and a Morrey type estimate for quasiminimizers. Combining this with Ekeland's variational principle, we obtain local Hölder continuity for $ω$-minimizers.
We prove higher integrability of the spatial gradient of weak solutions to parabolic systems with $ϕ$-growth, where $φ=φ(t)$ is a general Orlicz function. The parabolic systems need be neither degenerate nor singular. Our result is a generalized version of the one of J. Kinnunen and J. Lewis [Duke Math. J. 102 (2000), no. 2, 253--271] for the parabolic $p$-Laplace systems.
We prove higher integrability for the gradient of local minimizers of the Mumford-Shah energy functional, providing a positive answer to a conjecture of De Giorgi.
The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
本文证明了具有广义 Orlicz 增长的椭圆泛函的无界极小元属于 De Giorgi 类，并在此基础上建立了依赖于 $L^s$-范数的弱 Harnack 不等式，为多种泛函的 Harnack 型估计提供了更广泛的解释。
A way to measure the lower growth rate of $φ:Ω\times [0,\infty) \to [0,\infty)$ is to require $t \mapsto φ(x,t)t^{-r}$ to be increasing in $(0,\infty)$. If this condition holds with $r=1$, then \[ \inf_{u\in f+W^{1, φ}_0(Ω)}\int_Ωφ(x, |\nabla u|) \, dx \] with boundary values $f\in W^{1,φ}(Ω)$ does not necessary have a minimizer. However, if $φ$ is replaced by $φ^p$, then the growth condition holds with $r=p > 1$ and thus (under some additional conditions) the corresponding energy integral has a minimizer. We show that a sequence $(u_p)$ of such minimizers convergences when $p \to 1^+$ in a su…
Partial regularity for local minimizers of functionals with double-phase Orlicz growth
本文证明了双相Orlicz增长非自治泛函局部极小值的部分正则性，在连续模间隙条件和最优正则性假设下，将单相结果推广至双相情形。
Lipschitz regularity of weakly coupled vectorial almost-minimizers for Alt-Caffarelli functionals in Orlicz spaces
本文研究了弱耦合向量近极小解在Orlicz空间中Alt-Caffarelli泛函的Lipschitz正则性，提出了一种新的分析方法并提供了最佳局部Lipschitz正则性的证明。
We review some classical results and more recent insights about the regularity theory for local minimizers of the Mumford and Shah energy and their connections with the Mumford and Shah conjecture. We discuss in details the links among the latter, the porosity of the jump set and the higher integrability of the approximate gradient. In particular, higher integrability turns out to be related with an explicit estimate on the Hausdorff dimension of the singular set and an energetic characterization of the conjecture itself.
Higher integrability for parabolic PDEs with generalized Orlicz growth
针对广义Orlicz增长的抛物型系统，通过(A1)条件统一处理奇异与退化情形，证明梯度的高可积性，涵盖p增长、变指数和双相增长等特例。
Under appropriate assumptions on the $N(Ω)$-fucntion, the De Giorgi process is presented in the framework of Musielak-Orlicz-Sobolev space to prove the Hölder continuity of fully nonlinear elliptic problems. As the applications, the Hölder continuity of the minimizers for a class of the energy functionals in Musielak-Orlicz-Sobolev spaces is proved; and furthermore, the Hölder continuity of the weak solutions for a class of fully nonlinear elliptic equations is provided.
We study the qualitative properties of functions belonging to the corresponding De Giorgi classes \begin{equation*} \int\limits_{B_{r(1-σ)}(x_{0})}\,\varPhi(x, |\nabla(u-k)_{\pm}|)\,dx \leqslant γ\,\int\limits_{B_{r}(x_{0})}\,\varPhi\bigg(x, \frac{(u-k)_{\pm}}{σr}\bigg)\,dx, \end{equation*} where $σ$, $r \in (0,1)$, $k\in \mathbb{R}$ and the function $\varPhi$ satisfies the non-logarithmic condition \begin{equation*} \bigg(r^{-n}\int\limits_{B_{r}(x_{0})}[\varPhi\big(x,\frac{v}{r}\big)]^{s}\,dx\bigg)^{\frac{1}{s}}\bigg(r^{-n}\int\limits_{B_{r}(x_{0})}[\varPhi\big(x,\frac{v}{r}\big)]^{-t}\,dx\b…
Good access to the crack and integrability of the full gradient for Griffith almost-minimizers in the plane
研究平面上Griffith能量几乎极小值函数，通过证明裂纹补集被有限John域覆盖及位移满足估计，推导全梯度可积性与迹线有限性，得出二维中此类函数局部属于SBV^2空间的结论。
Self-improving property for certain degenerate functionals with generalized Orlicz growth
本文研究了在特定增长条件下，变分积分的自改进性质及最小值的存在性。
A two-phase problem with degenerate operator in Orlicz-Sobolev spaces
本文研究Orlicz-Sobolev空间中带$\Phi$-Laplacian算子的两相问题，证明了能量泛函极小元的存在性、有界性和Log-Lipschitz正则性，并指出其自由边界具有有限周长。
We prove higher summability for the gradient of minimizers of strongly convex integral functionals of the Calculus of Variations with (p,q)-Growth conditions in low dimension. Our procedure is set in the framework of Fractional Sobolev Spaces and renders the desired regularity as the result of an approximation technique relying on estimates obtained through a careful use of difference quotients.
In the first part of this article we deal with the existence of at least three non-trivial weak solutions of a nonlocal problem with nonstandard growth involving a nonlocal Robin type boundary condition. The second part of the article is devoted to study eigenvalues and minimizers of several nonlocal problems for the fractional $g-$Laplacian $(-Δ_g)^s$ with different boundary conditions, namely, Dirichlet, Neumann and Robin.
Stochastic-Periodic Homogenization of a Class of Minimization Problems in Orlicz-Sobolev Spaces
本文在Orlicz-Sobolev空间中发展随机双尺度收敛方法，证明一类高振荡极小化问题的极小化子收敛到均匀化问题的极小化子。
The aim of this paper is to study the heterogeneous optimization problem \begin{align*} \mathcal {J}(u)=\int_Ω(G(|\nabla u|)+qF(u^+)+hu+λ_{+}χ_{\{u>0\}} )\text{d}x\rightarrow\text{min}, \end{align*} in the class of functions $ W^{1,G}(Ω)$ with $ u-φ\in W^{1,G}_{0}(Ω)$, for a given function $φ$, where $W^{1,G}(Ω)$ is the class of weakly differentiable functions with $\int_ΩG(|\nabla u|)\text{d}x<\infty$. The functions $G$ and $F$ satisfy structural conditions of Lieberman's type that allow for a different behavior at $0$ and at $\infty$. {}{Moreover, $F$ allows for a subcritical growth.} Given …
Gradient integrability for bounded $\mathrm{BD}$-minimizers
本文证明了退化椭圆对称梯度泛函的有界松弛极小子具有弱梯度，并在已知最尖锐的椭圆性范围内实现了该结果，同时首次得出了 $\mathrm{BD}(Ω)$ 上面积型泛函极小子的 Sobolev 正则性。
In this manuscript we study the following optimization problem: given a bounded and regular domain $Ω\subset \mathbb{R}^N$ we look for an optimal shape for the "$\mathrm{W}-$vanishing window" on the boundary with prescribed measure over all admissible profiles in the framework of the Orlicz-Sobolev spaces associated to constant for the "Sobolev trace embedding". In this direction, we establish existence of minimizer profiles and optimal sets, as well as we obtain further properties for such extremals. Finally, we also place special emphasis on analyzing the corresponding optimization problem i…
Partial regularity for $\mathbb{A}$-quasiconvex functionals with Orlicz growth
针对基于椭圆算子的微分表达式的一类泛函，在Orlicz增长和自然强拟凸性条件下，通过约化为全梯度部分正则性情形，证明了极小元的部分正则性。
Stochastic two-scale convergence in the mean in Orlicz-Sobolev's spaces and applications to the homogenization of an integral functional
本文研究了 Orlicz-Sobolev 空间中具有凸且非标准增长被积函数的积分泛函的随机均质化，将均值双尺度收敛的经典紧性结果推广至该空间，并证明了一类高度振荡极小化问题序列的极小值点收敛于均质化问题的极小值点。
We study minimizers of the Dirichlet phi-energy integral with generalized Orlicz growth. We prove the Kellogg property, the set of irregular points has zero capacity, and give characterizations of semiregular boundary points. The results are new ever for the special cases double phase and Orlicz growth.
In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form \[ \mathfrak{F}( v, Ω)= \int_Ω \! F(x, Dv(x)) \, dx, \] where, for ${n > 2}$ and $N\ge 1$, $Ω$ is a bounded open set in $\mathbb{R}^n$, $u \in W^{1,1}(Ω, \mathbb{R}^N)$ and the energy density $F:Ω\times \mathbb{R}^{N \times n}\to \mathbb{R}$ satisfies the so called variable growth conditions. The main novelty of the paper is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the $x$ variable.
We prove the higher integrability of the gradient for minimizers of the thermal insulation problem, an analogue of De Giorgi's conjecture for the Mumford-Shah functional. We deduce that the singular part of the free boundary has Hausdorff dimension strictly less than $n-1$.
Regularity improvement for the minimizers of the two-dimensional Griffith energy
本文证明二维Griffith能量连通极小化子的奇异集维数小于1，并提高对称梯度的可积性。
Calderon-Zygmund estimates for generalized double phase equations with matrix weights
本文研究了具有Orlicz增长和可变矩阵权重的广义双相方程的Calderon-Zygmund估计，通过非均匀椭圆双相结构与满足小log-BMO条件的退化或奇异矩阵权重的结合，证明了弱解加权梯度的更高积分性，扩展了双相问题和加权椭圆方程的Calderon-Zygmund理论。
We consider the optimization problem of minimizing $\int_ΩG(|\nabla u|)+λχ_{\{u>0\}} dx$ in the class of functions $W^{1,G}(Ω)$ with $u-ϕ_0\in W_0^{1,G}(Ω)$, for a given $ϕ_0\geq 0$ and bounded. $W^{1,G}(Ω)$ is the class of weakly differentiable functions with $\int_ΩG(|\nabla u|) dx<\infty$. The conditions on the function G allow for a different behavior at 0 and at $\infty$. We prove that every solution u is locally Lipschitz continuous, that they are solution to a free boundary problem and that the free boundary, $\partial\{u>0\}\cap Ω$, is a regular surface. Also, we introduce the notion o…
Regularity results for Hölder minimizers to functionals with non-standard growth
本文研究非自治泛函在$(p,q)$-增长和Besov系数假设下Hölder连续极小元的正则性，证明了其更高可积性和可微性。
Surface-to-Volume Homogenization of NonLinear Steklov Eigencouples with NonHomogeneous Orlicz Growth in Perforated Domains
该研究针对穿孔域中具有非齐次Orlicz增长的非线性Steklov特征对，通过强逼近微观迹的方法，建立了其均匀化极限的相关结果。
Distortion of Hausdorff measures under Orlicz--Sobolev maps
本文系统研究了 Orlicz-Sobolev 映射对零或有限 Hausdorff 测度子集的影响，用 Young 函数给出规范函数畸变的显式公式，并发现新现象、恢复并强化了经典 Sobolev 空间结果。
Orlicz space relaxation of total variation denoising
本文提出一种基于Orlicz空间松弛的全变分去噪模型，通过对数密度校正减少阶梯效应，并证明其收敛到ROF模型。
We study the uniqueness of generalized $p$-minimal surfaces in the Heisenberg group. The generalized $p$-area of a graph defined by $u$ reads $\int |\nabla u+\vec{F}| + Hu$. If $u$ and $v$ are two minimizers for the generalized $p$-area satisfying the same Dirichlet boundary condition, then we can only get $N_{\vec{F}}(u)$ $=$ $N_{\vec{F}}(v)$ (on the nonsingular set) where $N_{\vec{F}}(w)$ $:=$ $\frac{\nabla w+\vec{F}}{|\nabla w+\vec{F}|}.$ To conclude $u$ $=$ $v$ (or $\nabla u$ $=$ $\nabla v)$, it is not straightforward as in the Riemannian case, but requires some special argument in general…
Parabolic De Giorgi classes with doubly nonlinear, nonstandard growth: local boundedness under exact integrability assumptions
该研究定义了含非平衡能量估计的抛物型De Giorgi函数类$\mathcal{PDG}$，证明其在奇异抛物算子的三类增长条件下局部有界，次临界情形的结果为全新成果，无需额外可积性条件。
We consider a class of integral functionals with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space W^{1;q}. We prove a result of higer differentiability for the minimizers. We also infer a result of Lipschitz regularity of minimizers if q > n, and a result of higher integrability for the gradient if q = n. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.
We establish local $C^{1,α}$-regularity for some $α\in(0,1)$ and $C^α$-regularity for any $α\in(0,1)$ of local minimizers of the functional \[ v\ \mapsto\ \int_Ωϕ(x,|Dv|)\,dx, \] where $ϕ$ satisfies a $(p,q)$-growth condition. Establishing such a regularity theory with sharp, general conditions has been an open problem since the 1980s. In contrast to previous results, we formulate the continuity requirement on $ϕ$ in terms of a single condition for the map $(x,t)\mapsto ϕ(x,t)$, rather than separately in the $x$- and $t$-directions. Thus we can obtain regularity results for functionals without…
On singular problems in nonreflexive fractional Orlicz-Sobolev spaces
研究非反射性分数Orlicz-Sobolev空间中奇异拟线性问题正解的存在性和唯一性，提出新的测试函数构建方法以克服能量函数泛化困难。
We establish local higher integrability and differentiability results for minimizers of variational integrals $$ \mathfrak{F}(v,Ω) = \int_Ω /! F(Dv(x)) \, dx $$ over $W^{1,p}$--Sobolev mappings $u \colon Ω\subset {\mathbb R}^n \to {\mathbb R}^N$ satisfying a Dirichlet boundary condition. The integrands $F$ are assumed to be autonomous, convex and of $(p,q)$ growth, but are otherwise not subjected to any further structure conditions, and we consider exponents in the range $1<p \leq q < p^{\ast}$, where $p^{\ast}$ denotes the Sobolev conjugate exponent of $p$.
Under appropriate assumptions on the $N(Ω)$-fucntion, the $L^{Φ(A(x,\cdot))}$ estimate for the gradient of the minimizers of a class of energy functional in Musielak-Orlicz-Sobolev space $W^{1,A(x,\cdot)}$ is presented by using Calderón-Zygmund decomposition.
Lipschitz regularity of almost minimizers in a Bernoulli problem with non-standard growth
本文研究了具有非标准增长的单相 Bernoulli 型泛函非负几乎极小子的最优 Lipschitz 正则性，并在 Orlicz 模背景下通过 Campanato 型估计提出了新的 Höder 正则性刻画。
Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane
本文证明平面中Griffith断裂能极小化子及几乎极小化子的裂纹集一致可修正，进而得出奇异集维数严格小于1，并证明梯度更高可积性，肯定回答De Giorgi猜想类似物。
Fractional higher differentiability of solutions to strongly nonlinear Stokes systems
本文研究强非线性斯托克斯系统的解的分数正则性，利用奥里奇和奥里奇-索波列夫空间分析，证明对称梯度和压力的分数索波列夫正则性。
We mainly discuss superquadratic minimization problems for splitting-type variational integrals on a bounded Lipschitz domain $Ω\subset \mathbb{R}^2$ and prove higher integrability of the gradient up to the boundary by incorporating an appropriate weight-function measuring the distance of the solution to the boundary data. As a corollary, the local Hölder coefficient with respect to some improved Hölder continuity is quantified in terms of the function ${\rm dist}(\cdot,\partial Ω)$. The results are extended to anisotropic problems without splitting structure under natural growth and elliptici…
We pursue the study of a model convex functional with orthotropic structure and nonstandard growth conditions, this time focusing on the sub-quadratic case. We prove that bounded local minimizers are locally Lipschitz. No restriction on the ratio between the highest and the lowest growth rates are needed. The result holds also in presence of a non-autonomous lower order term, under sharp integrability assumptions. Finally, we prove higher differentiability of bounded local minimizers, as well.
Higher integrability for variational integrals with non-standard growth
本文研究满足受控 $(p,q)$-增长的自主变分泛函，在放宽至 $q/p<1+2/(n-1)$ 的条件下证明极小元的更高梯度可积性与部分正则性。
Strong existence for free-discontinuity problems with non-standard growth
该研究针对广义 Orlicz 增长下的自由间断能量，在 $SBV^φ$ 空间中证明 Ahlfors 型正则性，从而拓展非标准增长自由间断问题极小元的正则性理论。
Partial regularity and higher integrability for A-quasiconvex variational problems
本文研究了变分问题中最小解的部分连续性及更高积分性，考虑了p增长问题和线性常秩PDE算子，以及Dirichlet边界条件下的A-自由场。
We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and Hölder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is Hölder continuous.
Regularity for vectorial double phase functionals with two modulating coefficients and logarithmic-type growth
本文研究具有两个调制系数和对数型增长的向量值双相泛函，证明在系数满足适当连续条件下极小元及其梯度的局部Hölder正则性，方法结合Gehring型高阶可积性与比较论证。
Rectifiability of free boundaries in singular diffusion problems
针对带有奇异反应项的退化扩散泛函的极小元，证明自由边界是(n-1)-可求长的，方法基于由逐点梯度估计导出的可积性及零点集的部分Hausdorff维数估计。
Lipschitz regularity for orthotropic functionals with general growth
研究一类具有φ增长的退化正交各向异性泛函局部极小值的局部利普希茨正则性，克服相关欧拉-拉格朗日方程椭圆性在各坐标方向单独退化及缺乏代数尺度不变性等困难，用正则化近似等方法证明其局部利普希茨连续。
Revisiting multi-phase variational problems: A Muckenhoupt weight approach
本文研究多相能量泛函局部极小值的正则性，通过引入Muckenhoupt权重替代经典Hölder连续性假设，建立高阶可积性、局部有界性和Hölder连续性，并首次证明多相设置下的Harnack不等式。
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