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Dissipative and Non-Dissipative Evolutionary Quasi-Variational Inequalities with Gradient Constraints

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Abstract

Evolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A semi-discretization in time is employed for the study of the problems and the derivation of a numerical solution scheme. Convergence of the discretization procedure is proven and properties of the original infinite dimensional problem, such as existence, extra regularity and non-decrease in time, are derived. The proposed numerical solver reduces to a finite number of gradient-constrained convex optimization problems which can be solved rather efficiently. The paper ends with a report on numerical tests obtained by a variable splitting algorithm involving different nonlinearities and types of constraints.

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References

  1. Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued Laplace Transforms and Cauchy Problems birkhäuser (2010)

  2. Azevedo, A., Miranda, F., Santos, L.: Variational and quasivariational inequalities with first order constraints. J. Math. Anal. Appl. 397(2), 738–756 (2013). https://doi.org/10.1016/j.jmaa.2012.07.033

    Article  MathSciNet  MATH  Google Scholar 

  3. Azevedo, A., Miranda, F., Santos, L.: Stationary Quasivariational Inequalities with Gradient Constraint and Nonhomogeneous Boundary Conditions, pp. 95–112. Springer, Berlin (2014)

    MATH  Google Scholar 

  4. Baiocchi, C., Capelo, A.: Variational and quasivariational inequalities Wiley-Interscience (1984)

  5. Barret, J.W., Prigozhin, L.: A quasi-variational inequality problem in superconductivity. Mathematical Models and Methods in Applied Sciences 20(5), 679–706 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barrett, J.W., Prigozhin, L.: A quasi-variational inequality problem arising in the modeling of growing sandpiles. ESAIM Math. Model. Numer. Anal. 47(4), 1133–1165 (2013). https://doi.org/10.1051/m2an/2012062

    Article  MathSciNet  MATH  Google Scholar 

  7. Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011). https://doi.org/10.1007/978-1-4419-9467-7. With a foreword by Hédy Attouch

    Google Scholar 

  8. Bensoussan, A., Lions, J.L.: Controle impulsionnel et inéquations quasi-variationnelles d’évolutions. C. R. Acad. Sci. Paris 276, 1333–1338 (1974)

    MATH  Google Scholar 

  9. Beremlijski, P., Haslinger, J., Kocvara, M., Outrata, J.: Shape optimization in contact problems with Coulomb friction. SIAM J. Optim. 13, 561–587 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brézis, H., Sibony, M.: Equivalence de deux inéquations variationnelles et applications. Archive Rat. Mech. Anal. 41, 254–265 (1971)

    Article  MATH  Google Scholar 

  11. Brézis, H., Stampacchia, G.: Sur la régularité de la solution d’inéquations elliptiques. Bulletin de la S. M. F. 96, 153–180 (1968)

    MATH  Google Scholar 

  12. Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  13. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: Introduction to nonlinear analysis: Applications Kluwer (2003)

  14. DiBenedetto, E.: Real analysis. Advanced Texts Series. Birkhauser, Cambridge (2002). http://books.google.at/books?id=5ddbKSkaL8EC

    Google Scholar 

  15. Duvaut, G., Lions, J.P.: Les inéquations en mécanique et en physique. Dunod, Paris (1972)

    MATH  Google Scholar 

  16. Fattorini, H.O.: Infinite dimensional optimization and control theory. Cambridge university press, Cambridge (1999)

    Book  MATH  Google Scholar 

  17. Fukao, T., Kenmochi, N.: Abstract theory of variational inequalities with Lagrange multipliers and application to nonlinear PDEs. Math. Bohem. 139(2), 391–399 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Fukao, T., Kenmochi, N.: Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete Contin. Dyn. Syst. 35(6), 2523–2538 (2015). https://doi.org/10.3934/dcds.2015.35.2523

    Article  MathSciNet  MATH  Google Scholar 

  19. Glowinski, R., Lions, J.P., Trémolières, R.: Numerical analysis of variational inequalities North-Holland (1981)

  20. Harker, P.T.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)

    Article  MATH  Google Scholar 

  21. Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups American Mathematical Society (1957)

  22. Hintermüller, M., Kopacka, I.: Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm. SIAM J. Optim. 20(2), 868–902 (2009). https://doi.org/10.1137/080720681

    Article  MathSciNet  MATH  Google Scholar 

  23. Hintermüller, M., Rasch, J.: Several path-following methods for a class of gradient constrained variational inequalities. Comput. Math. Appl. 69(10), 1045–1067 (2015). https://doi.org/10.1016/j.camwa.2014.12.001

    Article  MathSciNet  Google Scholar 

  24. Hintermüller, M., Rautenberg, C.N.: A sequential minimization technique for elliptic quasi-variational inequalities with gradient constraints. SIAM J. Optim. 22(4), 1224–1257 (2012). https://doi.org/10.1137/110837048

    Article  MathSciNet  MATH  Google Scholar 

  25. Hintermüller, M., Rautenberg, C.N.: Parabolic quasi-variational inequalities with gradient-type constraints. SIAM J. Optim. 23(4), 2090–2123 (2013). https://doi.org/10.1137/120874308

    Article  MathSciNet  MATH  Google Scholar 

  26. Hintermüller, M., Rautenberg, C.N.: On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities. Port. Math. 74(1), 1–35 (2017). https://doi.org/10.4171/PM/1991

    Article  MathSciNet  MATH  Google Scholar 

  27. Hintermüller, M., Rautenberg, C.N., Rösel, S.: Density of convex intersections and applications. In: Proceedings of the Royal Society of London A: Mathematical Physical and Engineering Sciences 473(2205) (2017)

  28. Kadoya, A., Kenmochi, N., Niezgódka, M.: Quasi-variational inequalities in economic growth models with technological development. Adv. Math. Sci. Appl. 24(1), 185–214 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Kano, R., Kenmochi, N., Murase, Y.: Parabolic quasi-variational inequalities with non-local constraints. Adv. Math. Sci. Appl. 19(2), 565–583 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Kano, R., Murase, Y., Kenmochi, N.: Nonlinear evolution equations generated by subdifferentials with nonlocal constraints. In: Nonlocal and abstract parabolic equations and their applications, Banach Center Publ., vol. 86, pp. 175–194. Polish Acad. Sci. Inst. Math., Warsaw (2009). https://doi.org/10.4064/bc86-0-11

  31. Kenmochi, N.: Parabolic quasi-variational diffusion problems with gradient constraints. Discrete Contin. Dyn. Syst. Ser. S 6(2), 423–438 (2013). https://doi.org/10.3934/dcdss.2013.6.423

    Article  MathSciNet  MATH  Google Scholar 

  32. Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. SIAM (2000)

  33. Kleinhans, M.G., Markies, H., de Vet, S.J., in ’t Veld, A.C., Postema, F.N.: Static and dynamic angles of repose in loose granular materials under reduced gravity. Journal of Geophysical Research: Planets 116(E11), n/a–n/a (2011). https://doi.org/10.1029/2011JE003865

    Article  Google Scholar 

  34. Kravchuk, A.S., Neittaanmäki, P.J.: Variational and Quasi-variational Inequalities in Mechanics. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  35. Kunze, M., Rodrigues, J.: An elliptic quasi-variational inequality with gradient constraints and some of its applications. Mathematical Methods in the Applied Sciences 23, 897–908 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lions, J.L.: Sur le côntrole optimal des systemes distribuées. Enseigne 19, 125–166 (1973)

    MATH  Google Scholar 

  37. Lions, J.L.: Asymptotic behaviour of solutions of variational inequalitites with highly oscillating coefficients. Applications of Methods of Functional Analysis to Problems in Mechanics. In: Proceedings of the Joint Symposium IUTAM/IMU. Lecture Notes in Mathematics, p 503. Springer, Berlin (1975)

  38. Miranda, F., Rodrigues, J.F., Santos, L.: A class of stationary nonlinear maxwell systems. Mathematical Models and Methods in Applied Sciences 19(10), 1883–1905 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Miranda, F., Rodrigues, J.F., Santos, L.: On a p-curl system arising in electromagnetism. Discrete Contin. Dyn. Syst. Ser. S 5(3), 605–629 (2012)

    MathSciNet  MATH  Google Scholar 

  40. Mosco, U.: Convergence of convex setis and solutions of variational inequalities. Adv. Math. 3(4), 510–585 (1969)

    Article  MATH  Google Scholar 

  41. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 3, 373–375 (2009)

    Article  MATH  Google Scholar 

  42. Prigozhin, L.: Quasivariational inequality describing the shape of a poured pile. Zhurnal Vichislitel’noy Matematiki i Matematicheskoy Fiziki 7, 1072–1080 (1986)

    MathSciNet  Google Scholar 

  43. Prigozhin, L.: Sandpiles and river networks: extended systems with non-local interactions. Phys. Rev. E 49, 1161–1167 (1994)

    Article  MathSciNet  Google Scholar 

  44. Prigozhin, L.: On the Bean critical-state model in superconductivity. Eur. J. Appl. Math. 7, 237–247 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  45. Prigozhin, L.: Sandpiles, river networks, and type-ii superconductors. Free Boundary Problems News 10, 2–4 (1996)

    Google Scholar 

  46. Prigozhin, L.: Variational model of sandpile growth. Euro. J. Appl. Math. 7, 225–236 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rodrigues, J.F.: Obstacle problems in mathematical physics North-Holland (1987)

  48. Rodrigues, J.F., Santos, L.: A parabolic quasi-variational inequality arising in a superconductivity model. Ann. Scuola Norm. Sup. Pisa Cl. Sci. XXIX, 153–169 (2000)

    MathSciNet  MATH  Google Scholar 

  49. Rodrigues, J.F., Santos, L.: Quasivariational solutions for first order quasilinear equations with gradient constraint. Arch. Ration. Mech. Anal. 205(2), 493–514 (2012). https://doi.org/10.1007/s00205-012-0511-x

    Article  MathSciNet  MATH  Google Scholar 

  50. Showalter, R.E.: Monotone operators in banach space and nonlinear partial differential equations american mathematical society (1997)

  51. Simon, J.: Compact sets in the space L p(0,T; B). Annali di Matematica pure ed applicata CXLVI(IV), 65–96 (1987)

    MATH  Google Scholar 

  52. Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  53. Verfürth, R.: A posteriori error estimation techniques for finite element methods, Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013). https://doi.org/10.1093/acprof:oso/9780199679423.001.0001

    Book  MATH  Google Scholar 

  54. Willett, D., Wong, J.: On the discrete analogues of some generalizations of Gronwall’s inequality. Monatshefte für Mathematik 69(4), 362–367 (1965). https://doi.org/10.1007/BF01297622

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

Authors

Corresponding author

Correspondence to C. N. Rautenberg.

Additional information

This work was carried out in the framework of the DFG under grant no. HI 1466/7-1 “Free Boundary Problems and Level Set Methods”, the DFG-funded SFB-TRR 154 Subproject B02, as well as the Research Center Matheon supported by the Einstein Foundation Berlin within projects OT1, SE5, and SE15/19. The authors further gratefully acknowledge the support of the DFG through the DFG-SPP 1962: Priority Programme Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization within projects 10, 11 and 13.

Appendices

Appendix A: Lower Solutions for VIs

The following result is due (to the best of our knowledge) to Bensoussan and is included for the sake of completeness.

Proposition 5

Let\(A:{H_{0}^{1}}({\Omega })\to H^{-1}({\Omega })\)belinear, bounded and uniformly monotone. Additionally, suppose thatif\(v\in {H_{0}^{1}}({\Omega })\),thenAv,v+〉≤ 0.LetφiL(Ω) withi = 1, 2 be such 0 ≤ φ1φ2a.e.,

$$\mathbf{K}(\varphi_{i}):=\{v\in {H_{0}^{1}}({\Omega}): v \leq \varphi_{i} \text{a.e. }\}, $$

and suppose that fiL2(Ω) with i = 1, 2 and f1f2a.e.. Then, y1y2a.e., where\(y_{i}=\mathbb {S}(A, f_{i}, \mathbf {K}(\varphi _{i}))\).

Further, let\(\varphi \in L_{+}^{\infty }({\Omega })\)and fL2(Ω). If zK(φ) satisfies

$$ \langle Az-f,\phi\rangle \leq 0, \quad \forall \phi\in {H_{0}^{1}}({\Omega}): \quad \phi\geq 0 \text{ a.e.}, $$
(64)

we say z is a lower solution for the triple (A, f, K(φ)). For any lower solution z, we have that\(z\leq \mathbb {S}(A, f, \mathbf {K}(\varphi ))\).

Proof

Since yiK(φi) for i = 1, 2, then 0 ≤ v1 := min(y1,y2) = y1 − (y1y2)+ψ1 and 0 ≤ v2 := max(y1,y2) = y2 + (y1y2)+ψ2. Hence, from 〈Ayifi,viyi〉≥ 0 for i = 1, 2, we obtain

$$ \langle Ay_{1}-f_{1},-(y_{1}-y_{2})^{+}\rangle\geq 0, \quad \text{ and } \quad \langle Ay_{2}-f_{2},-(y_{1}-y_{2})^{+}\rangle\leq 0. $$
(65)

Subtracting the second inequality from the first one, we observe

$$\langle A(y_{1}-y_{2}),(y_{1}-y_{2})^{+}\rangle\leq (f_{1}-f_{2},(y_{1}-y_{2})^{+})\leq 0, $$

since f1f2 ≤ 0. Since A is uniformly monotone and 〈Av,v+〉≤ 0, for all \(v\in {H_{0}^{1}}({\Omega })\), we obtain the following chain of inequalities:

$$\begin{array}{@{}rcl@{}} c|(y_{1}-y_{2})^{+}|^{2}_{{H_{0}^{1}}({\Omega})} &\leq& \langle A(y_{1}-y_{2})^{+}, (y_{1}-y_{2})^{+}\rangle\\ &\leq&\langle A(y_{1}-y_{2})^{+}, (y_{1}-y_{2})^{+}\rangle- \langle A(y_{1}-y_{2})^{-}, (y_{1}-y_{2})^{+}\rangle\\ &=&\langle A(y_{1}-y_{2}), (y_{1}-y_{2})^{+}\rangle\leq 0. \end{array} $$

Therefore, (y1y2)+ = 0 a.e., that is, y1y2 a.e. in Ω.

Let \(y= \mathbb {S}(A, f, \varphi )\), so yK(φ) and

$$ \langle Ay-f,v-y\rangle\geq 0, \quad \forall y\in \mathbf{K}(\varphi). $$
(66)

Replacing v = yϕ with \( \phi \in {H_{0}^{1}}({\Omega })\) and ϕ ≥ 0 a.e. in Ω, we observe that \(y= \mathbb {S}(A, f, \mathbf {K}(\varphi ))\) is a lower solution for the triple (A,f,K(φ)). Now we prove that if z is an arbitrary lower solution, the zy a.e. in Ω. Let ϕ = (zy)+ and v = max(y,z) = y + (zy)+ on (64) and (66), respectively, then

$$\langle Az-f,-(z-y)^{+}\rangle\geq 0 \quad \text{ and } \quad \langle Ay-f,-(z-y)^{+}\rangle\leq 0. $$

These are exactly the same inequalities as in (65). Therefore, we have that (zy)+ = 0, i.e., zy a.e. in Ω. □

Appendix B: Proof of Lemma 1

Proof

Consider first a and i = 1. Let \(w\in \mathscr{K}({\Psi })\) and note that the condition “\(w^{N}(t)\in \mathbf {K}({\Phi }(t_{n-1}^{N},u_{n-1}^{N}))\) with \(t\in [t_{n-1}^{N},{t_{n}^{N}})\)” is equivalent to

$$ |\nabla w^{N}(\tau)|\leq {\sum}_{m = 1}^{N} {\Phi}(t_{m-1}^{N}, u_{-}^{N}(\tau))\chi_{[t_{m-1}^{N},{t_{m}^{N}})}(\tau)=:\hat{{\Phi}}(\tau, u_{-}^{N}(\tau)), \quad \tau\in [0, T]. $$
(67)

Denote by \(\{\tilde {u}^{N}\}\) the convergent subsequence obtained in Theorem 2, i.e., \(\tilde {u}^{N}\rightarrow u^{*}\) in C([0,T]; L2(Ω)). Then, by the inequality in (19) we also have that

$$\lim_{N\rightarrow\infty} |u_{-}^{N}- u^{*}|_{L^{\infty}(0, T;L^{2}({\Omega}))}=\lim_{N\rightarrow\infty} \sup_{t\in [0,T]}|u_{-}^{N}(t)- u^{*}(t)|_{L^{2}({\Omega})}= 0. $$

By Assumption 1, we have that Φ : [0,T] × L2(Ω) → L(Ω) is uniformly continuous, i.e., for any 𝜖 > 0, there exists δ(𝜖) > 0 such that

$$|t_{1}-t_{2}|+|y_{1}-y_{2}|_{L^{2}({\Omega})}<\delta(\epsilon) \quad \Longrightarrow\quad|{\Phi}(t_{1}, y_{1})-{\Phi}(t_{2}, y_{2})|_{L^{\infty}({\Omega})}<\epsilon. $$

Therefore, for sufficiently large N we have that

$$\frac{1}{N}+ |u_{-}^{N}- u^{*}|_{C([0, T];L^{2}({\Omega}))}<\delta(\epsilon) \quad \Longrightarrow\quad|\hat{{\Phi}}(\tau, u_{-}^{N}(\tau))-{\Phi}(\tau, u^{*}(\tau))|_{L^{\infty}({\Omega})}<\epsilon, $$

for all τ ∈ [0,T]. Recall that by assumption we have that the mapping Φ satisfies: 1) Φ(t,v) ≥ ν > 0 a.e. in Ω, for a.e. t ∈ [0,T] and all vL2(Ω). 2) It is non-decreasing in both variables. 3) T↦Φ(T,v) maps bounded sets in L2(Ω) into bounded sets in L(Ω). Then, we define \(\varphi _{N}(t,x):=\hat {{\Phi }}(t, u_{-}^{N}(t))(x)\) and φ(t,x) := Φ(t,u(t))(x) with (t,x) ∈ Q := [0,T] ×Ω. It follows that φN,φL(Q) and also

$$ \varphi_{N},\varphi\geq \nu>0 :\quad \varphi_{N}\rightarrow\varphi \text{ in } L^{\infty}(Q), \text{ as } N\rightarrow\infty. $$
(5)

Now, we prove that for any η ∈ (0, 1), there is an N(η) such that

$$0\leq \eta \varphi (z)\leq \varphi_{N}(z) \quad \text{ a.e. } z\in Q, $$

for NN(η) . In fact, let η ∈ (0, 1) be arbitrary, and consider the sets

$$Q_{N}:=\{z\in Q: \eta \varphi (z)>\varphi_{N}(z) \text{ a.e.}\}. $$

Then, for almost all zQN, we have

$$|\varphi - \varphi_{N}|_{L^{\infty}(Q)}\geq \varphi (z)- \varphi_{N}(z)>(1-\eta)\varphi (z)\geq (1-\eta)\nu>0. $$

But since \(|\varphi - \varphi _{N}|_{L^{\infty }(Q)}\rightarrow 0\), there exists \(N(\eta )\in \mathbb {N}\), such that |QN| = 0 for all NN(η).

Let {ηj} be a monotonically increasing sequence in (0,1) such that \(\lim _{j\rightarrow \infty }\eta _{j}= 1\). Let \(w\in L^{2}(0, T; {H_{0}^{1}}({\Omega }))\) satisfy |∇w(t)(x)|≤ φ(t,x). Then, wj := ηjw fulfils

$$|\nabla w^{j}(t)(x)|\leq \eta_{j} |\nabla w(t)(x)|\leq \eta_{i} \varphi(t,x) \leq \varphi_{N(\eta_{j})}(t, x), $$

for almost all (t,x) ∈ Q. Finally, \(|w^{j}-w|_{L^{2}(0, T; {H_{0}^{1}}({\Omega }))}=(1-\eta _{j})|w|_{L^{2}(0, T; {H_{0}^{1}}({\Omega }))}\leq (1-\eta _{j})|\varphi |_{L^{\infty }(Q)} \rightarrow 0\) as j. This proves the statement concerning \(w\in \mathscr{K}({\Psi })\).

Next, we focus on a and i = 2. For the same sequence {ηj} as before, suppose \(w\in L^{2}(0, T; {H_{0}^{1}}({\Omega }))\) is arbitrary and such that \(w\in \mathscr{K}^{\pm }({\Psi })\). Then wj(t) = ηjw(t) belongs to \(\mathbf {K}^{\pm }(\varphi _{N(\eta _{j})}(t,\cdot ))\), i.e., for d(x) := dist(x,Ω)

$$-\varphi_{N(\eta_{j})}d(x) \leq- \eta_{j}\varphi d(x) \leq \eta_{j}w \leq \eta_{j}\varphi d(x)\leq \varphi_{N(\eta_{j})}d(x), $$

(where we have omitted “(t, x)” for the sake of brevity) for a.e. t ∈ (0,T),x ∈Ω. Further, it follows that \(|w^{j}-w|_{L^{2}(0, T; {H_{0}^{1}}({\Omega }))}=(1-\eta _{j})|w|_{L^{2}(0, T; {H_{0}^{1}}({\Omega }))}\rightarrow 0\) as j, and hence proves this case i = 2 for the a statement and an analogous argument can be used to prove i = 3.

We now consider b. Since \(\tau \in [t_{n-1}^{N}, {t_{n}^{N}})\) is constant, \(\lim _{N\rightarrow \infty }t_{n-1}^{N}=\tau \) and \(\phi ^{N}={\Phi }(t^{N}_{n-1}, u^{N}_{n-1})={\Phi }(t^{N}_{n-1}, \tilde {u}^{N}(t^{N}_{n-1}))\). By (22) we have ϕNϕ in L(Ω) and in addition ϕN,ϕν > 0 a.e. in Ω. These are the conditions in (68) (with Q exchanged by Ω), and using the same argument we can prove that given a monotonically increasing sequence {ηj} in (0,1) with \(\lim _{j\rightarrow \infty }\eta _{j}= 1\), then wj = ηjw satisfies \(w^{j}\in \mathbf {K}(\phi ^{N(\eta _{j})})\), provided that wK(ϕ), and wjw in \({H_{0}^{1}}({\Omega })\). This proves the i = 1 case and analogous modifications of the argument in a. proves the cases concerning i = 2 and i = 3. □

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Hintermüller, M., Rautenberg, C.N. & Strogies, N. Dissipative and Non-Dissipative Evolutionary Quasi-Variational Inequalities with Gradient Constraints. Set-Valued Var. Anal 27, 433–468 (2019). https://doi.org/10.1007/s11228-018-0489-0

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