Existence of minimizers in Evans chapter 8
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I am trying to understand the following point in Evans PDE book. In chapter 8, page 465, he proved the existence of minimizer to the following problem, let $U$ be open, bounded with connected, $C^1$ boundary and consider
$$ inf_{win mathcal{A}} int_U L(Dw(x),w(x),x);dx = inf_{win mathcal{A}} J[w]$$
over
$$mathcal{A} = {win W^{1,q}(U): w = g;text{on};partial U}$$.
Assuming coercivity of $L$, that is
$$L(p,z,x) geq alpha |p|^q - beta$$
for some $alpha > 0$ and $betageq 0$, then if we let
$$ m = inf_{win mathcal{A}} int_U L(Dw(x),w(x),x);dx$$
and take a minimizing sequence $u_kin mathcal{A}$ such that
$$ J(u_k) longrightarrow m$$
as $klongrightarrow infty$, one can easily see that
$$ Vert Du_kVert_{L^q(U)} leq C$$
He then use the fact that $u_k = g$ on $partial U$ to conclude that
$$ Vert u_kVert_{L^q(U)} leq C'$$
as well, which I don't understand how? It is of course clear if $g = 0$ by using Poincare's inequality, but if $gneq 0$ then I don't know how to do that estimate.
pde sobolev-spaces calculus-of-variations
asked Dec 9 '18 at 2:17
SeanSean
532513
$endgroup$
add a comment |
$begingroup$
I am trying to understand the following point in Evans PDE book. In chapter 8, page 465, he proved the existence of minimizer to the following problem, let $U$ be open, bounded with connected, $C^1$ boundary and consider
$$ inf_{win mathcal{A}} int_U L(Dw(x),w(x),x);dx = inf_{win mathcal{A}} J[w]$$
over
$$mathcal{A} = {win W^{1,q}(U): w = g;text{on};partial U}$$.
Assuming coercivity of $L$, that is
$$L(p,z,x) geq alpha |p|^q - beta$$
for some $alpha > 0$ and $betageq 0$, then if we let
$$ m = inf_{win mathcal{A}} int_U L(Dw(x),w(x),x);dx$$
and take a minimizing sequence $u_kin mathcal{A}$ such that
$$ J(u_k) longrightarrow m$$
as $klongrightarrow infty$, one can easily see that
$$ Vert Du_kVert_{L^q(U)} leq C$$
He then use the fact that $u_k = g$ on $partial U$ to conclude that
$$ Vert u_kVert_{L^q(U)} leq C'$$
as well, which I don't understand how? It is of course clear if $g = 0$ by using Poincare's inequality, but if $gneq 0$ then I don't know how to do that estimate.
pde sobolev-spaces calculus-of-variations
asked Dec 9 '18 at 2:17
SeanSean
532513
$endgroup$
2
$begingroup$
I think you can just consider $v_k=u_k-w$, where $win W^{1,q}(U)$ is a fixed function with boundary value $g$.
$endgroup$
– MaoWao
Dec 9 '18 at 14:55
add a comment |
$begingroup$
I am trying to understand the following point in Evans PDE book. In chapter 8, page 465, he proved the existence of minimizer to the following problem, let $U$ be open, bounded with connected, $C^1$ boundary and consider
$$ inf_{win mathcal{A}} int_U L(Dw(x),w(x),x);dx = inf_{win mathcal{A}} J[w]$$
over
$$mathcal{A} = {win W^{1,q}(U): w = g;text{on};partial U}$$.
Assuming coercivity of $L$, that is
$$L(p,z,x) geq alpha |p|^q - beta$$
for some $alpha > 0$ and $betageq 0$, then if we let
$$ m = inf_{win mathcal{A}} int_U L(Dw(x),w(x),x);dx$$
and take a minimizing sequence $u_kin mathcal{A}$ such that
$$ J(u_k) longrightarrow m$$
as $klongrightarrow infty$, one can easily see that
$$ Vert Du_kVert_{L^q(U)} leq C$$
He then use the fact that $u_k = g$ on $partial U$ to conclude that
$$ Vert u_kVert_{L^q(U)} leq C'$$
as well, which I don't understand how? It is of course clear if $g = 0$ by using Poincare's inequality, but if $gneq 0$ then I don't know how to do that estimate.
pde sobolev-spaces calculus-of-variations
asked Dec 9 '18 at 2:17
SeanSean
532513
$endgroup$
I am trying to understand the following point in Evans PDE book. In chapter 8, page 465, he proved the existence of minimizer to the following problem, let $U$ be open, bounded with connected, $C^1$ boundary and consider
$$ inf_{win mathcal{A}} int_U L(Dw(x),w(x),x);dx = inf_{win mathcal{A}} J[w]$$
over
$$mathcal{A} = {win W^{1,q}(U): w = g;text{on};partial U}$$.
Assuming coercivity of $L$, that is
$$L(p,z,x) geq alpha |p|^q - beta$$
for some $alpha > 0$ and $betageq 0$, then if we let
$$ m = inf_{win mathcal{A}} int_U L(Dw(x),w(x),x);dx$$
and take a minimizing sequence $u_kin mathcal{A}$ such that
$$ J(u_k) longrightarrow m$$
as $klongrightarrow infty$, one can easily see that
$$ Vert Du_kVert_{L^q(U)} leq C$$
He then use the fact that $u_k = g$ on $partial U$ to conclude that
$$ Vert u_kVert_{L^q(U)} leq C'$$
as well, which I don't understand how? It is of course clear if $g = 0$ by using Poincare's inequality, but if $gneq 0$ then I don't know how to do that estimate.
pde sobolev-spaces calculus-of-variations
pde sobolev-spaces calculus-of-variations
asked Dec 9 '18 at 2:17
SeanSean
532513
asked Dec 9 '18 at 2:17
SeanSean
532513
asked Dec 9 '18 at 2:17
SeanSean
532513
asked Dec 9 '18 at 2:17
SeanSean
532513
asked Dec 9 '18 at 2:17
SeanSean
532513
532513
2
$begingroup$
I think you can just consider $v_k=u_k-w$, where $win W^{1,q}(U)$ is a fixed function with boundary value $g$.
$endgroup$
– MaoWao
Dec 9 '18 at 14:55
add a comment |
2
$begingroup$
I think you can just consider $v_k=u_k-w$, where $win W^{1,q}(U)$ is a fixed function with boundary value $g$.
$endgroup$
– MaoWao
Dec 9 '18 at 14:55
2
2
$begingroup$
I think you can just consider $v_k=u_k-w$, where $win W^{1,q}(U)$ is a fixed function with boundary value $g$.
$endgroup$
– MaoWao
Dec 9 '18 at 14:55
$begingroup$
I think you can just consider $v_k=u_k-w$, where $win W^{1,q}(U)$ is a fixed function with boundary value $g$.
$endgroup$
– MaoWao
Dec 9 '18 at 14:55
add a comment |
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$begingroup$
I think you can just consider $v_k=u_k-w$, where $win W^{1,q}(U)$ is a fixed function with boundary value $g$.
$endgroup$
– MaoWao
Dec 9 '18 at 14:55